Integrals

Previous Years' CBSE Board Questions

7.2 Integration as an Inverse Process of Differentiation

MCQ

<math>\int \frac{\sec x}{\sec x - \tan x} dx</math> equals

secx - tanx + c

1. (b) secx + tanx + c
2. (d) <math>-(secx + tanx) + c</math>

(2023)

<math>tanx - secx + c</math>

VSA (1 mark)

Find: <math display="block">\int \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} dx</math> (Al 2017)

Write the antiderivative of <math>\left(3\sqrt{x} + \frac{1}{\sqrt{x}}\right)</math> (Delhi 2014)

Evaluate: ∫cos^-1(sinx)dx (Delhi 2014)

1. Evaluate: <math display="block">\int \frac{dx}{\sin^2 x \cos^2 x}</math>

(Foreign 2014, Delhi 2014 C) [II]

1. Find: <math display="block">\int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx</math> (Delhi 2014 C)

SAI (2 marks)

Find: <math>\int \sqrt{1-\sin 2x} dx</math>, <math>\frac{\pi}{4} < x < \frac{\pi}{2}</math> (Delhi 2019)

1. Evaluate: <math>\int \frac{\cos 2x + 2\sin^2 x}{\cos^2 x} dx</math>

7.3 Methods of Integration

MCQ

1. <math>e^{\operatorname{Slog} x} dx</math> is equal to
2. <math>\frac{x^5}{5} + C</math> (b) <math>\frac{x^6}{6} + C</math>
3. <math>5x^4 + C</math> (d) <math>6x^5 + C</math> (2020)

1. <math>\int x^2 e^{x^3} dx</math> equals
2. <math>\frac{1}{3}e^{x^3} + C</math>
3. <math>\frac{1}{2}e^{x^4} + C</math>

(Delhi 2019) (Ap)

1. <math>\frac{1}{2}e^{x^2} + C</math> (2020)

1. <math>\frac{1}{2}e^{x^3} + C</math>
2. <math display="block">\int \frac{e^x(1+x)}{\cos^2(xe^x)} dx</math> is equal to

<math>\cot(xe^x) + c</math>

VSA (1 mark)

1. Find: <math>\int \frac{2x}{\sqrt[3]{x^2+1}} dx</math> (2020)
2. Find: <math display="block">\int \frac{\sin^6 x}{\cos^8 x} dx</math> (AI 2014 C)

(2 marks)

1. Find: <math display="block">\int \frac{dx}{\sqrt{4x-x^2}}</math> (Term II, 2021-22) Ev

SA II (3 marks)

1. Find: <math display="block">\int \frac{\sin x}{\sin(x-2a)} dx</math> (Term II, 2021-22 C)

16. Find: <math>\int \frac{1}{e^x + 1} dx</math> (Term II, 2021-22)

17. Find: <math>\int \frac{2x}{(x^2+1)(x^2+2)} dx</math> (Term II, 2021-22)

(4 marks)

1. Integrate the function <math>\frac{\cos(x+a)}{\sin(x+b)}</math> with respect to x. (Al 2019)
2. Find <math display="block">\int \frac{(3\sin\theta-2)\cos\theta}{5-\cos^2\theta-4\sin\theta} d\theta.</math> (Delhi 2016)
3. Evaluate: <math display="block">\int \frac{\sin(x-a)}{\sin(x+a)} dx</math> (Foreign 2015)

(2018) (2018) 21. Evaluate: <math>\int \frac{\sin^6 x + \cos^6 x}{\sin^2 x + \cos^2 x} dx</math> (Delhi 2014)

7.4 Integrals of Some Particular Functions

VSA (1 mark)

22. Find: <math>\int \frac{dx}{\sqrt{9-4x^2}}</math>

(2020)

23. Find: <math display="block">\int \frac{dx}{9 + 4 \sqrt{2}}</math>

SAI (2 marks)

1. Find: <math display="block">\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} dx</math>

1. Find: <math>\int \frac{dx}{\sqrt{5-4x-2x^2}}</math>

(Al 2019)

1. Find: <math>\int \frac{dx}{x^2 + 4x + 8}</math>

(Delhi 2017)

1. tan (xex) + c (b)
2. <math>\cot(e^x) + c</math> (d) <math>\tan [e^x (1+x)] + c</math> (Al 2017) (2020) (40)

1. Find: <math>\int \frac{dx}{5-8x-x^2}</math>

SA II (3 marks) (Al 2017)

1. Find: <math>\int \frac{e^x}{\sqrt{5-4e^x-e^{2x}}} dx</math>

LAI (4 marks)

(Foreign 2016) [Ap]

(Delhi 2016) Ap (Delhi 2015)

1. Find <math>\int \frac{\sqrt{x}}{\sqrt{3}} dx</math>.
2. Find <math>\int (\sqrt{\tan x} + \sqrt{\cot x}) dx</math>. (Al 2015, 2014)

(Delhi 2015C)

1. Evaluate: <math display="block">\int \frac{x+2}{2x^2+6x+5} dx</math>
2. Find <math>\int \frac{x+3}{\sqrt{5-4y-2y^2}} dx</math>.
3. Evaluate: <math display="block">\int \frac{x+2}{\sqrt{x^2+6x+4}} dx</math> (Al 2014)
4. Evaluate: <math>\int \frac{5x-2}{1+2x+2x^2} dx</math> (Delhi 2014C) (An

LA II (5/6 marks)

1. Evaluate: <math display="block">\int \frac{1}{\cos^4 x + \sin^4 x} dx</math> (Al 2014)
2. Evaluate: <math display="block">\int \frac{1}{\sin^4 x + \sin^2 x \cos^2 x + \cos^4 x} dx</math>

7.5 Integration by Partial Fractions

SAI (2 marks)

1. Find <math>\int \frac{x+1}{(x+2)(x+3)} dx</math>.

SAII (3 marks) (2020) (An

38. Find: <math>\int \frac{x^2}{x^2+6x+12} dx</math>

LAI (4 marks)

39. Find: <math>\int \frac{x^2}{(x^2+1)(3x^2+4)} dx</math>

(2020C)

1. Find: <math>\int \frac{x^3+1}{x^3-x} dx</math>
2. Find: <math>\int \frac{3x+5}{x^2+3x+19} dx</math>

(Delhi 2019) (A) SAI (2 marks)

1. Evaluate: <math>\int \frac{x^2 + x + 1}{(x^2 + 1)(x + 2)} dx</math>

(Term II, 2021-22) (An)

(Delhi 2019)

1. Find: <math>\int \frac{2\cos x}{(1-\sin x)(1+\sin^2 x)} dx</math>
2. Find: <math>\int \frac{2x}{(x^2+1)(x^2+2)^2} dx</math>.
3. Find: <math display="block">\int \frac{\cos \theta}{(4+\sin^2 \theta)(5-4\cos^2 \theta)} d\theta</math>
4. Find: <math>\int \frac{x^2}{4 + x^2} dx</math>

(Al 2016)

1. Find: <math display="block">\int \frac{(x^2+1)(x^2+4)}{(x^2+3)(x^2-5)} dx</math>
2. Find: \( \int \frac{dx}{\sin x + \sin 2x} \)
3. Evaluate: <math display="block">\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} dx</math> (Foreign 2015)

(AI 2015C) 50. Find: <math>\int \frac{x}{(x-1)^2(x+2)} dx</math> (Delhi 2015C)

1. Find: <math>\int \frac{x}{(x^2+1)(x-1)} dx</math> (AI 2015C)

52. Evaluate: <math>\int \frac{x^2}{(x^2 + 1)(x^2 + 4)} dx</math> (Delhi 2014C) (A)

1. Find: <math>\int \frac{x^3}{x^4 + 2x^2 + 3} dx</math> (AI 2014C)

LA II (5/6 marks)

1. Find <math>\int \frac{x^2 + x + 1}{(x + 1)^2 (x + 2)} dx</math>. (Delhi 2014C)

(AI 2014) An 7.6 Integration by Parts

MCQ

55. <math>\int e^x \left(\frac{x \log x + 1}{x}\right) dx</math> is equal to

(a) <math>\log (e^x \log x) + c</math> (b) <math>\frac{e^x}{x} + c</math> (c) <math>x \log x + e^x + c</math> (d) <math>e^x \log x + c</math>

(2023) 56. <math>\int \frac{e^x}{x+1} [1+(x+1)\log(x+1)] dx</math> equals (a) <math>\frac{e^x}{v+1} + c</math> (b) <math>e^x \frac{x}{x+1} + c</math>

(Term II, 2021-22) (c) <math>e^x \log(x+1) + e^x + c</math>

1. <math>e^x \log(x+1) + c</math>

(2020) An VSA (1 mark)

1. Find: <math>\int x^4 \log x \, dx</math> (2020)

(2018) 59. Find: <math>\int \sin^{-1}(2x) dx</math>

(Delhi 2017) 60. Find: <math>\int x \cdot \tan^{-1} x \, dx</math> (AI 2019) (EV)