# CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Additional Exercise Textbook Exercise questions and Answers.

## CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 1.
∫$$\sqrt{1-\sin 2 x}$$ dx
Solution:
I = ∫$$\sqrt{1-\sin 2 x}$$ dx
= ∫$$\sqrt{(\cos x-\sin x)^2}$$ dx
= ∫(cos x – sin x) dx
= sin x + cos x + c

Question 2.
∫$$\frac{d x}{1+\sin x}$$
Solution:
I = ∫$$\frac{d x}{1+\sin x}$$
= ∫$$\frac{1-\sin x}{\cos ^2 x}$$
= ∫sec2 x – sec x tan x dx
= tan x – sec x + c

Question 3.
∫$$\frac{\sin x}{1+\sin x}$$ dx
Solution:

Question 4.
∫$$\frac{\sec x}{\sec x+\tan x}$$ dx
Solution:
I = ∫$$\frac{\sec x}{\sec x+\tan x}$$ dx
= ∫$$\frac{\sec x(\sec x-\tan x)}{\sec ^2 x-\tan ^2 x}$$ dx
= ∫sec2 x – sec x tan x dx
= tan x – sec x + c

Question 5.
∫$$\frac{1+\sin x}{1-\sin x}$$ dx
Solution:
I = ∫$$\frac{1+\sin x}{1-\sin x}$$ dx
= ∫$$\frac{(1+\sin x)^2}{\cos ^2 x}$$ dx
= ∫[sec2 x+ tan2 x+ 2sec x tan x) dx
= ∫[2sec2 x – 1 + 2sec x tan x) dx
= 2tan x – x + 2sec x + c

Question 6.
∫tan-1 (sec x + tan x) dx
Solution:

Question 7.
∫$$\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}$$ dx
Solution:
I = ∫$$\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}$$ dx
= ∫$$\frac{\left(2 \cos ^2 x-1\right)-\left(2 \cos ^2 \alpha-1\right)}{\cos x-\cos \alpha}$$ dx
= 2 ∫(cos x + cos α) dx
= 2 sin x + 2x cos α + c

Question 8.
∫tan-1$$\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}$$ dx
Solution:

Question 9.
∫$$\frac{d x}{\sqrt{x+1+} \sqrt{x+2}}$$
Solution:

Question 10.
∫$$\frac{2+3 x}{3-2 x}$$ dx
Solution:

Question 11.
∫$$\frac{d x}{\sqrt{x}+x}$$
Solution:

Question 12.
∫$$\frac{d x}{1+\tan x}$$
Solution:

Question 13.
∫$$\frac{x+\sqrt{x+1}}{x+2}$$ dx (Hints put : $$\sqrt{x+1}$$ = t)
Solution:

Question 14.
∫sin-1$$\sqrt{\frac{x}{a+x}}$$ dx (Hints put : x = a tan2 t)
Solution:

Question 15.
∫ex$$\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right)$$ dx
Solution:

Question 16.
∫$$\frac{\left(x^2+1\right) e^x}{(x+1)^2}$$ dx
Solution:

Question 17.
∫$$\frac{x^2-1}{x^4+x^2+1}$$ dx
Solution:

Question 18.
∫$$\frac{x^2 d x}{x^4+x^2+1}$$
Solution:

Question 19.
∫$$\sqrt{\cot x}$$ dx
Solution:

Question 20.
∫$$(\sqrt{\tan x}+\sqrt{\cot x})$$ dx
Solution:

Question 21.
∫$$\frac{\mathrm{dx}}{x\left(x^4+1\right)}$$
Solution:

Question 22.
∫$$\frac{\mathrm{dx}}{e^x-1}$$
Solution:

Question 23.
∫$$\frac{(x-1)(x-2)(x-3)}{(x+4)(x-5)(x-6)}$$ dx
Solution:

Question 24.
∫$$\frac{d x}{\left(e^x-1\right)^2}$$
Solution:

Question 25.
∫$$\frac{d x}{\sin x \cos ^2 x}$$
Solution:

Question 26.
$$\int_2^4 \frac{\left(x^2+x\right) d x}{\sqrt{2 x+1}}$$
Solution:

Question 27.
$$\int_{-a}^a \sqrt{\frac{a-x}{a+x}}$$ dx
Solution:

Let a2 – x2 = t2
-2x dx = 2t dt
x = -a ⇒ 0 t = 0
x = a ⇒ t = 0
= 0
I = aI1 – I2 = aπ

Question 28.
$$\int_0^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x})$$ dx
Solution:

Question 29.
$$\int_0^{\pi / 2} \frac{\cos x d x}{1+\cos x+\sin x}$$
Solution:

Question 30.
$$\int_0^1$$x (1 – x)n dx
Solution:

Question 31.
$$\int_0^{\pi / 2}$$sin 2x log (tan x) dx
Solution:

Question 32.
$$\int_0^{\pi / 2} \frac{\sin ^2 x d x}{\sin x+\cos x}$$
Solution:

Question 33.
$$\int_0^{\pi / 2} \frac{\sin ^2 x d x}{1+\sin x \cos x}$$
Solution:

Question 34.
$$\int_0^{\pi / 2} \frac{x d x}{\sin x+\cos x}$$
Solution:

Question 35.
Prove that $$\int_0^\pi$$ x sin3 x dx = $$\frac{2 \pi}{3}$$
Solution:

Question 36.
$$\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x d x}{\sin x+\cos x}$$
Solution:

Question 37.
$$\int_0^\pi$$|cos x| dx
Solution:

Question 38.
$$\int_1^4$$(|x – 1| + |x – 2| + |x – 3|) dx
Solution:

Question 39.
$$\int_{-\pi / 2}^{\pi / 2}$$(sin |x| + cos |x|) dx
Solution:

Question 40.
$$\int_0^\pi$$log (1 + cos x) dx
Solution: