CBSE Class 12 Maths Chapter 1: Integrals NCERT Solutions

Reviewed by · Updated on

NCERT Solutions PDF Class 12 PDF

This chapter provides essential NCERT Solutions for Class 12 Mathematics, focusing on Chapter 1: Integrals. It covers fundamental integration formulas, including power rule, logarithmic, exponential, trigonometric, inverse trigonometric, and special forms. The solutions explain how to find antiderivatives using the inspection method and various standard integration formulas. Key topics include integrating functions like \sin 2x, \cos 2x, e^{2x}, ax+b, \sin^2 2x, \cos^2 2x, \sin 3x, \cos 3x, \sin(2x+5), \cos(2x+5), \sin^2(2x+5), \cos^2(2x+5), \tan^2 x, \cot^2 x, \sin 10x, \cos 7x, \frac{1}{1+x^2}, \frac{1}{\sqrt{1-x^2}}, \frac{1}{x^2-1}, \frac{1}{2-x^2}, \frac{1}{\sqrt{2-x^2}}, \frac{1}{\sqrt{2x^2+1}}, \frac{1}{5x^2+4}, \frac{1}{\sqrt{16-9x^2}}, \frac{1}{1+4x^2}, \frac{1}{\sqrt{x^2+4x+6}}, \frac{1}{\sqrt{5-4x-2x^2}}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}{2x^2+x-1}, \frac{1}{6x+5-2x^2}, \frac{1}{x^2+2x+2}, \frac{1}

Home

Quick info

BoardCBSE
ClassClass 12
Subjectगणित-II
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 1

PDF preview

Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.

Loading document …
Page of
Loading page …

Questions and Solutions

अध्याय-7

समाकलन

(Integrals)

(Important Formulae and Definitions)

1. <math display="block">\int x^n dx = \frac{x^{n+1}}{n+1} + c, (n \neq -1)</math>

2. <math>\int \frac{1}{x} dx = \log_e |x| + c</math>

<math display="block">4. \int a^x dx = \frac{a^x}{\log_e a} + c</math>

<math display="block">3. \quad \int e^x dx = e^x + c</math>

<math display="block">5. \int \sin x \, dx = -\cos x + c</math>

<math display="block">\mathbf{6.} \int \cos x \, dx = \sin x + c</math>

7. <math display="block">\int \sec^2 x \, dx = \tan x + c</math>

<math display="block">8. \int \csc^2 x \ dx = -\cot x + c</math>

9. <math>\int \sec x \tan x dx = \sec x + c</math>

10. <math>\int \csc x \cot x \, dx = -\csc x + c</math>

11. <math display="block">\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + c</math>

12. <math>\int \frac{1}{\sqrt{1+x^2}} dx = \tan^{-1} x + c</math>

13. <math display="block">\int \frac{1}{x\sqrt{x^2-1}} dx = \sec^{-1} x + c</math>

14. <math display="block">\int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c</math>

15. <math>\int \frac{dx}{ax+b} = \frac{1}{a} \log|ax+b| + c</math>

16. <math>\int e^{ax} dx = \frac{1}{a} e^{ax} + c</math>

18. <math display="block">\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \frac{x}{a} + c</math>

17. <math display="block">\int \frac{dx}{a^2 + r^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + c</math>

19. <math display="block">\int \frac{dx}{x\sqrt{x^2 - a^2}} dx = \frac{1}{a} \sec^{-1} \frac{x}{a} + c</math>

346

  1. <math display="block">\int \tan x \, dx = -\log|\cos x| + c \, \text{ an log } |\sec x| + c</math>

<math display="block">21. \int \cot x \, dx = \log|\sin x| + c</math>

  1. <math>\int \csc x \, dx = \log \left| \tan \frac{x}{2} \right| + c = \log \left| \csc x - \cot x \right| + c</math>
  2. <math>\int \sec x \, dx = \log |\sec x + \tan x| + c = \log \left|\tan \left(\frac{\pi}{4} + \frac{x}{2}\right)\right| + c</math>

24. <math>\int e^{ax} \sin bx \ dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + c</math>

  1. <math>\int e^{ax} \cos bx \ dx = \frac{e^{ax}}{a^2 + b^2} (b \sin bx + a \cos bx) + c</math>

26. <math>\int \sqrt{a^2 - x^2} dx = \frac{1}{2} \left| x \sqrt{a^2 - x^2} + a^2 \sin^{-1} \frac{x}{a} \right| + c</math>

  1. <math display="block">\int \sqrt{a^2 + x^2} dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \log_e |x + \sqrt{a^2 + x^2}| + c</math>
  2. <math display="block">\int \sqrt{x^2 - a^2} dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}| + c</math>
  3. <math>\int \frac{1}{x^2 - a^2} dx = \frac{1}{2a} \log_e \left| \frac{x - a}{x + a} \right| + c</math>, <math>\forall a \in A</math>
  4. <math>\int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log_e \left| \frac{x+a}{x-a} \right| + c</math>, <math>\forall a \in A</math>
  1. <math display="block">\int \frac{dx}{ax^2 + bx + c} = \frac{2}{\sqrt{4ac - b^2}} \tan^{-1} \frac{2ax + b}{\sqrt{4ac - b^2}}, \text{ so } 4ac > b^2</math> <math display="block">= -\frac{1}{\sqrt{b^2 - 4ac}} \log_e \frac{2ax + b - \sqrt{b^2 - 4ac}}{2ax + b + \sqrt{b^2 - 4ac}}, \text{ जब } 4ac < b^2.</math>

प्रश्नावली 7·1

निम्नलिखित फलनों के प्रतिअवकलज (समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए :

प्रश्न 1. sin 2x.

हल: हम जानते हैं कि

<math display="block">\frac{d}{dx}\cos 2x = -2\sin 2x</math>

<math display="block">-\frac{1}{2}\frac{d}{dx}\cos 2x = \sin 2x</math>

उत्तर

<math display="block">\int \sin 2x \, dx = -\frac{1}{2} \cos 2x + C.</math>

Important links

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.