## Rajasthan Board RBSE Class 12 Maths Chapter 9 Integration Ex 9.6

**Exalnate the following :**

Question 1.

(a) ∫ x cos x dx

(b) ∫ x sec^{2} x dx

Solution :

Question 2.

(a) ∫ x^{3} e^{-x} dx

(b) ∫ x^{3} sin x dx

Solution:

Question 3.

(a) ∫ x^{3} (log x)^{2} dx

(b) ∫ x^{3} ex^{2} 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 – 4x^{3}) dx

Now, let x = sin t, then dx = cos t dt

∴ I = ∫ sin^{-1} (3 sin t – 4 sin^{3} 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) sec^{2} x dx

Solution:

∫e^{x} (sin x + cos x) sec^{2} 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= ∫θ^{2} . cos θ dθ

= θ^{2} sin θ – ∫20 sin θ dθ + C

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