ML Aggarwal Class 12 Maths Solutions Section A Chapter 8 Integrals Ex 8.1

Effective ISC Mathematics Class 12 Solutions Chapter 8 Integrals Ex 8.1 can help bridge the gap between theory and application.

ML Aggarwal Class 12 Maths Solutions Section A Chapter 8 Integrals Ex 8.1

Very short answer type questions :

Evaluate the following integrals :

Question 1.
(i) ∫ x⁵ dx
(i) ∫ \(\frac{1}{x^3}\) dx
(iii) ∫ \(\frac{1}{\sqrt{y}}\) dy
Solution:
(i) ∫ x⁵ dx = \(\frac{x^{5+1}}{5+1}\) + C
[∵ ∫ x⁵ dx = \(\frac{x^{n+1}}{n+1}\) + C ; n ≠ 1]
= \(\frac{x^6}{6}\) + C

(ii) ∫ \(\frac{1}{x^3}\) dx
= ∫ x^{– 3} dx
= \(\frac{x^{-3+1}}{-3+1}\) + C
= \(\frac{x^{-2}}{-2}\) + C
= – \(\frac{1}{2 x^2}\) + C

(iii) ∫ \(\frac{1}{\sqrt{y}}\) dy
= ∫ y^{\(\frac{-1}{2}\)} dy
= \(\frac{y^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\) + C
= 2√y + C.

Question 2.
(i) ∫ \(\frac{1}{\sqrt[3]{x}}\) dx
(ii) ∫ sin² x cosec² x dx
(iii) ∫ sin x sec² x dx
Solution:
(i) ∫ \(\frac{1}{\sqrt[3]{x}}\) dx
= ∫ x^{– 1/3} dx
= \(\frac{x^{-\frac{1}{3}+1}}{-\frac{1}{3}+1}\) + C
= \(\frac{3}{2}\) x^2/3 + C

(ii) ∫ sin² x cosec² x dx
= ∫ sin² x × \(\frac{1}{\sin ^2 x}\) dx
= ∫ 1 dx = x + C

(iii) ∫ sin x sec² x dx
= ∫ \(\frac{\sin x}{\cos ^2 x}\) dx
= ∫ tan x sec x dx
= sec x + C
[∵ ∫ tan x sec x dx = sec x + C]

Question 3.
(i) ∫ 5^x dx
(ii) ∫ 7^x e^x dx
(iii) ∫ \(\frac{4^x}{9^x}\) dx
Solution:
(i) ∫ 5^x dx
= \(\frac{5^x}{\log _e 5}\) + C
[∵ ∫ a^x dx
= \(\frac{a^x}{\log _e a}\) + C

(ii) ∫ 7^x e^x dx
= ∫ (7e)^x dx
= \(\frac{(7 e)^x}{\log 7 e}\) + C

(iii) ∫ \(\frac{4^x}{9^x}\) dx
= ∫ \(\left(\frac{4}{9}\right)^x\) dx
= \(\frac{\left(\frac{4}{9}\right)^x}{\log \frac{4}{9}}\) dx + C

Question 4.
(i) ∫ \(\frac{\sin x}{1-\sin ^2 x}\) dx
(ii) ∫ \(\frac{\cos 2 x+2 \sin ^2 x}{\cos ^2 x}\) dx (NCERT)
(iii) ∫ \(\sqrt{\frac{1}{2}(1+\cos 2 x)}\) dx
Solution:
(i) ∫ \(\frac{\sin x}{1-\sin ^2 x}\) dx
= ∫ \(\frac{\sin x}{\cos ^2 x}\) dx
= ∫ tan x sec x dx
= sec x + C

(ii) ∫ \(\frac{\cos 2 x+2 \sin ^2 x}{\cos ^2 x}\) dx
= ∫ \(\frac{1-2 \sin ^2 x+2 \sin ^2 x}{\cos ^2 x}\) dx
[∵ cos 2θ = 1 – 2 sin² θ]
= ∫ \(\frac{1}{\cos ^2 x}\) dx
= ∫ sec² x dx = tan x
[∵ ∫ sec² x dx = tan x + C]

(iii) ∫ \(\sqrt{\frac{1}{2}(1+\cos 2 x)}\) dx
= ∫ \(\sqrt{\frac{1}{2} \times 2 \cos ^2 x}\) dx
= ∫ cos x dx
= sin x + C

Question 5.
(i) ∫ e^{2 log x} x^{– 3} dx
(ii) ∫ 5^{2 log₅ x} dx
(iii) ∫ 2^{log₄ x} dx
Solution:
(i) ∫ e^{2 log x} x^{– 3} dx
= ∫ e^{log x2} x^{– 3} dx
= ∫ x² . x^{– 3} dx
= ∫ x^{– 1} dx
= ∫ \(\frac{1}{x}\) dx
= log |x| + C
[∵ ∫ \(\frac{1}{x}\) dx = log |x| + C]

(ii) ∫ 5^{2 log₅ x} dx
= ∫ 5^{log₅ x2} dx
= ∫ x² dx
= \(\frac{x^{3}}{3}\) + C

(iii) ∫ 2^{log₄ x} dx
= ∫ 2^{\(\frac{\log x}{2 \log 2}\)} dx
= ∫ \(2^{\frac{1}{2} \log _2 x}\) dx
= ∫ 2^{log₂ x1/2} dx
= ∫ x^{\(\frac{1}{2}\)} dx
= \(\frac{x^{\frac{1}{2}+1}}{\left(\frac{1}{2}+1\right)}\) + C
= \(\frac{2}{3}\) x^{\(\frac{3}{2}\)} + C
[∵ ∫ xⁿ dx = \(\frac{x^{n+1}}{n+1}\) + C]