RBSE Solutions for Class 12 Maths Chapter 9 Integration Ex 9.6

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) ∫ e^2x e^ex 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.
∫ e^x (cot x + log sin x) dx
Solution:
Let I = ∫ e^x [log | sin x | + cot x] dx
= ∫ e^x log | sin x | dx + ∫ e^x cot x dx
= ∫ log | sin x | . e^x dx + ∫ e^x cot x dx
Now ∫ log | sin x | . e^x dx

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.
∫e^x [log (sec x + tan x) + sec x] dx
Solution:
∫e^x [log (sec x + tan x) + sec x] dx
= ∫e^x log (sec x + tan x) dx + ∫e^x sec x dx

Question 16.
∫e^x (sin x + cos x) sec² x dx
Solution:
∫e^x (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

RBSE Solutions for Class 12 Maths