# CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(a)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(a) Textbook Exercise Questions and Answers.

## CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(a)

Question 1.
(i) ∫2 dx
Solution:
∫2 dx = 2x + C

(ii) ∫3x2 dx
Solution:
∫3x2 dx = 3 $$\frac{x^3}{3}$$ + C = x3 + C

(iii) ∫4x3 dx
Solution:
∫4x3 dx = 4 $$\frac{x^4}{4}$$ + C = x4 + C

(iv) ∫x5 dx
Solution:
∫x5 dx = $$\frac{x^6}{6}$$ + C

(v) ∫x31 dx
Solution:
∫x31 dx = $$\frac{x^{32}}{32}$$ + C

(vi) ∫$$\left(2 \sqrt{x}+\frac{3}{\sqrt{x}}\right)$$ dx
Solution:

(vii) ∫$$\frac{1}{x \sqrt{x}}$$ dx
Solution:

(viii) ∫$$\left(x^{\frac{4}{7}}+\frac{1}{x^{\frac{1}{3}}}\right)$$ dx
Solution:

(ix) ∫$$\frac{4}{x}$$ dx
Solution:
∫$$\frac{4}{x}$$ dx = 4 ln |x| + C

(x) ∫$$\left(\frac{3}{x^2}+\frac{1}{x^{12}}\right)$$ dx
Solution:

(xi) ∫(x2 + √x)2 dx
Solution:

(xii) ∫(x + 3) (2 – x) dx
Solution:
∫(x + 3) (2 – x) dx
= ∫(2x + 6 – x2 – 3x) dx
= ∫(-x2 – x + 6) dx
= –$$\frac{1}{3}$$x3 – $$\frac{1}{2}$$x2 + 6x + C

(xiii) ∫$$\frac{(\sqrt{x}+2)^2}{x^4}$$ dx
Solution:

(xiv) ∫$$\frac{(x+\sqrt{x})(2 x+1)}{x^2}$$ dx
Solution:

Question 2.
(i) ∫cos dx
Solution:
∫cos dx = sin x + C

(ii) ∫$$\frac{d x}{\cos ^2 x}$$
Solution:
∫$$\frac{d x}{\cos ^2 x}$$ = ∫sec2 x dx = tan x + C

(iii) ∫$$\frac{d x}{1-\cos ^2 x}$$
Solution:
∫$$\frac{d x}{1-\cos ^2 x}$$ = ∫$$\frac{d x}{\sin ^2 x}$$
= ∫cosec2 x dx
= -cot x + C

(iv) ∫$$\frac{\sin x}{\cos ^2 x}$$ dx
Solution:
∫$$\frac{\sin x}{\cos ^2 x}$$ dx = ∫$$\frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}$$ dx
= ∫sec x . tan x dx = sec x + C

(v) ∫$$\frac{2 \cos x}{1-\cos ^2 x}$$ dx
Solution:

(vi) ∫$$\frac{1-\sin ^3 x}{\sin ^2 x}$$ dx
Solution:
∫$$\frac{1-\sin ^3 x}{\sin ^2 x}$$ dx = ∫(cosec2 x – sin x) dx
= -cot x + cos x + C

(vii) ∫$$\frac{\sin ^2 x}{1+\cos x}$$ dx
Solution:

(viii) ∫$$\frac{\cos 2 x}{\cos x+\sin x}$$ dx
Solution:

(ix) ∫$$\frac{\cos ^4 x-\sin ^4 x}{\cos x-\sin x}$$ dx
Solution:

(x) ∫$$\frac{\cos 2 x}{\sin ^2 x \cdot \cos ^2 x}$$ dx
Solution:

(xi) ∫$$\frac{a^2 \sin ^2 x+b^2 \cos ^2 x}{\sin ^2 2 x}$$ dx
Solution:

(xii) ∫(tan x + cot x)2 dx
Solution:

(xiii) ∫$$\frac{1-\cos 2 x}{1+\cos 2 x}$$ dx
Solution:

(xiv) ∫sec2 x . cosec2 x dx
Solution:

(xv) ∫$$\frac{\sin ^3 x+b \cos ^3 x}{\sin ^2 x \cos ^2 x}$$ dx
Solution:

(xvi) ∫$$\sqrt{1+\sin 2 x}$$ dx
Solution:

(xvii) ∫$$\sqrt{1-\cos 2 x}$$ dx
Solution:

(xviii) ∫$$\sqrt{1+\cos 2 x}$$ dx
Solution:
∫$$\sqrt{1+\cos 2 x}$$ dx = ∫√2 cos x dx
= √2 sin x + C

(xix) ∫$$\frac{\cos 3 x \cos 2 x+\sin 3 x \sin 2 x}{1-\cos ^2 x}$$ dx
Solution:

(xx) ∫(a cot x + b tan x)2 dx
Solution:
∫(a cot x + b tan x)2 dx
= ∫{a2 cot2 x + b2 tan2 x + 2ab cot x – tanx} xdx
= ∫(a2 cot2 x + b2 tan2 x + 2ab) dx
= a2 ∫cot2 x dx + b2 ∫tan2 x dx + 2ab ∫dx
= a2 ∫(cosec2 x – 1) dx + b2 ∫(sec2 x – 1) dx + 2ab x + C
= a2 (-cot x – x) + b2 (tan x – x) + 2ab x + C
= b2 tan x – a2 cot x – x (a2 + b2 – 2ab) + C
= b2 tan x – a2 cot x – (a – b)2 x + C
= b2 tan x – a2 cot x + C

Question 3.
(i) ∫(ex + 2) dx
Solution:
∫(ex +2) dx = ∫ex dx + 2 ∫dx
= ex + 2x + C

(ii) ∫3x dx
Solution:
∫3x dx = ∫$$\frac{3^x}{\ln 3}$$ + C

(iii) ∫ax+2 dx
Solution:
∫ax+2 dx = ∫ax a2 dx
= a2 . $$\frac{a^x}{\ln a}$$ + C
= $$\frac{a^{x+2}}{\ln a}$$ + C

(iv) ∫a3x dx
Solution:
∫a3x dx = $$\frac{1}{3} \frac{a^{3 x}}{\ln a}$$ + C

(v) ∫$$\frac{e^{2 x}+1}{e^x}$$ dx
Solution:
∫$$\frac{e^{2 x}+1}{e^x}$$ dx = ∫(ex + e-x) dx
= ex – e-x + C

Question 4.
(i) ∫$$\left(\frac{5}{\sqrt{1-x^2}}+\frac{7}{1+x^2}\right)$$ dx
Solution:

(ii) ∫$$\frac{3 x^2}{x^2+1}$$ dx
Solution:

(iii) ∫$$\frac{x^6}{x^2+1}$$ dx
Solution:

(iv) ∫$$\frac{x^4+x^3+x^2+x+2}{x^2+1}$$ dx
Solution:

(v) ∫$$\left(\sqrt{1-x^2}+\frac{x^2}{\sqrt{1-x^2}}\right)$$ dx
Solution:

(vi) ∫$$\frac{x^2+\sqrt{x^2+1}}{x^3 \sqrt{x^2-1}}$$ dx
Solution:

Question 5.
Find the unique antiderivative F(x) of f(x) = 2x2 + 1, whese F(o) = -2.
Solution:
F(x) = ∫f(x) dx = ∫(2x2 + 1) dx
= $$\frac{2}{3}$$x3 + x + C
But F(0) = -2
∴ -2 = C
Thus F(x) = $$\frac{2}{3}$$x3 + x – 2