NCERT Solutions for Class 12 Maths Chapter 7 Integrals — Exercise 7.20

Integrate the functions in Exercises 1 to 24

Question 1:3 1 x ✛ x

Question 2:1 x ✜ a xb ✜ ✜

Question 3:2 1 x ax x ✢ [Hint: Put x = a t ]

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Question 4:3 2 4 4 1 x x( 1 ✣ )

Question 5:1 1 2 3 1 x ✤ x [Hint: 1 1 1 1 2 3 3 6 1 1 x x x 1 x ✥ ✩ ✪ ✩ ✪ ✫ ✬ , put x = t 6 ]

Question 6:2 5 ( 1) ( 9) x x x

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Question 7:sin sin ( ) x x ✭ a

Question 8:5 log 4 log 3 log 2 log x x x x e e e e ✮ ✮

Question 9:2 cos 4 sin x ✛ x

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Question 10:8 8 2 2 sin cos 1 2sin cos x x x ✯ ✯

Question 11:1 cos ( ) cos ( ) x a x b

Question 12:3 8 1 x ✔ x

Question 13:(1 ) (2 ) x x x e ✰ e e ✰

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Question 14:2 2 1 ( 1 x x ✱ ) ( 4 ✱ )

Question 15:cos3 x e log sinx

Question 16:e 3 logx (x 4 + 1)– 1

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Question 17:f ✲ (ax + b) [f (ax + b)]n

Question 18:3 1 sin sin ( ) x x ✳

Question 19:1 1 1 1 sin cos sin cos x x x x ✴ ✴ ✴ ✴ ✛ ✵ , x ✶ [0, 1]

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Question 20:1 1 x x ✷ ✸

Question 21:2 sin 2 1 cos2 x x e x ✹ ✹

Question 22:2 2 1 ( 1) ( 2) x x x x

Question 23:– 1 1 tan 1 x x✁

Question 24:2 2 4 x 1 log ( 1) 2 log x x x Evaluate the definite integrals in Exercises 25 to 33

Question 25:2 1 sin 1 cos x x e dx x

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Question 26:4 4 4 0 sin cos cos sin x x dx x x ✓ ✔

Question 27:2 2 2 2 0 cos cos 4 sin x dx x x ✕ ✖ ✗

Question 28:3 6 sin cos sin 2 x x dx x ✘ ✘ ✚

Integrate the functions in Exercises 1 to 24.

Question 29:1 0 1 dx ✛ x ✜ x ✢

Question 30:4 0 sin cos 9 16 sin 2 x x dx x ✕ ✖ ✖ ✗

Question 31:2 1 0 sin 2 tan (sin ) x x dx ✣ ✤ ✥

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Question 32:0 tan sec tan x x dx x x

Question 33:4 1 [ 1| | 2 | | 3|] x ✩ ✪ x xd ✩ ✪ ✩ x ✫ Prove the following (Exercises 34 to 39)

Question 34:3 2 1 2 2 log ( 1) 3 3 dx x x ✬ ✭ ✭ ✮

Question 35:1 0 1 x x e dx ✯ ✫

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Question 36:1 17 4 1 x x cos 0 dx ✰ ✯ ✫

Question 37:3 2 0 2 sin 3 x dx ✱ ✲ ✢

Question 38:4 3 0 2 tan 1 log 2 x dx ✣ ✳ ✥

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Question 39:1 1 0 sin 1 2 x dx ✴ ✵ ✶ ✷ ✸

Question 40:Evaluate 1 2 3 0 x e dx ✰ ✫ as a limit of a sum. Choose the correct answers in Exercises 41 to 44

Question 41:x x dx e e✹ ✖ ✗ is equal to (A) tan–1 (e x ) + C (B) tan–1 (e –x) + C (C) log (e x – e –x) + C (D) log (e x + e –x) + C

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Question 42:2 cos2 (sin cos ) x dx x x ✥ is equal to (A) –1 C sin cos x x ✺ ✺ (B) log |sin cos | C x x ✖ ✖ (C) log |sin cos | C x x ✻ ✖ (D) 2 1 (sin cos ) x

Question 43:If f (a + b – x) = f (x), then ( ) b a x f x dx is equal to (A) ( ) 2 b a a b f b x dx (B) ( ) 2 b a a b f b x dx (C) ( ) 2 b a b a f x dx (D) ( ) 2 b a a b f x dx

Question 44:The value of 1 1 2 0 2 1 tan 1 x dx x x is (A) 1 (B) 0 (C) –1 (D) 4