Rajasthan Board RBSE Class 12 Maths Chapter 9 Integration Ex 9.6
Exalnate the following :
Question 1.
(a) ∫ x cos x dx
(b) ∫ x sec2 x dx
Solution :
Question 2.
(a) ∫ x3 e-x dx
(b) ∫ x3 sin x dx
Solution:
Question 3.
(a) ∫ x3 (log x)2 dx
(b) ∫ x3 ex2 dx
Solution:
Question 4.
(a) ∫ e2x eex dx
(b) ∫ (log x)2 dx
Solution:
Question 5.
Solution:
Question 6.
Solution:
(a) Let I = ∫ sin-1 (3x – 4x3) dx
Now, let x = sin t, then dx = cos t dt
∴ I = ∫ sin-1 (3 sin t – 4 sin3 t).cos t dt
= ∫ sin-1 (sin 31) cos t dt
= ∫ 3t cos t dt
= ∫ 3t cos t dt
I II
Question 7.
Solution:
Question 8.
Solution:
Question 9.
Solution:
Question 10.
Solution:
Question 11.
∫ ex (cot x + log sin x) dx
Solution:
Let I = ∫ ex [log | sin x | + cot x] dx
= ∫ ex log | sin x | dx + ∫ ex cot x dx
= ∫ log | sin x | . ex dx + ∫ ex cot x dx
Now ∫ log | sin x | . ex dx
Question 12.
Solution:
Question 13.
Solution:
Question 14.
Solution:
Question 15.
∫ex [log (sec x + tan x) + sec x] dx
Solution:
∫ex [log (sec x + tan x) + sec x] dx
= ∫ex log (sec x + tan x) dx + ∫ex sec x dx
Question 16.
∫ex (sin x + cos x) sec2 x dx
Solution:
∫ex (sin x + cos x) sec2 x dx
Question 17.
Solution:
Question 18.
Solution:
Question 19.
Solution:
Question 20.
Solution:
Question 21.
Solution:
Question 22.
∫(sin-1 x)2 dx
Solution:
Let I = ∫ (sin-1 x)2 dx
Let sin-1 x = θ
⇒ x = sin θ
∴ dx – cos θ dθ∴
∴ I= ∫θ2 . cos θ dθ
= θ2 sin θ – ∫20 sin θ dθ + C
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