Exercise 11.1 - Chapter 11 Integral Calculus 11th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Integral Calculus
Ex 11.1
Integrate the following with respect to ' $x$ ':
Question 1 .
(i) $\mathrm{x}^{11}$
(ii) $\frac{1}{x^7}$
(iii) $\sqrt[3]{x^4}$
(iv) $\left(x^5\right)^{\frac{1}{8}}$
Solution:
(i) $\int x^{11} d x=\frac{x^{12}}{12}+c$
(ii) $\int \frac{1}{x^7} d x=\int x^{-7} d x=\frac{x^{-7+1}}{-7+1}=\frac{-1}{6 x^6}+c$
(iii) $\int \sqrt[3]{x^4} d x=\int x^{4 / 3} d x=\frac{x^{4 / 3+1}}{\frac{4}{3}+1}=\frac{3}{7} x^{7 / 3}+c$
(iv) $\int\left(x^5\right)^{1 / 8} d x=\int x^{5 / 8} d x=\frac{8}{13} x^{13 / 8}+c$
Question 2 .
(i) $\frac{1}{\sin ^2 x}$
(ii) $\frac{\tan x}{\cos x}$
(iii) $\frac{\cos x}{\sin ^2 x}$
(iv) $\frac{1}{\cos ^2 x}$

Solution:
(i) $\int \frac{1}{\sin ^2 x} d x=\int \operatorname{cosec}^2 x d x=-\cot x+c$
(ii) $\int \frac{\tan x}{\cos x} d x=\int \sec x \tan x d x=\sec x+c$
(iii) $\int \frac{\cos x}{\sin ^2 x} d x=\int \operatorname{cosec} x \cot x d x=-\operatorname{cosec} x+c$
(iv) $\int \frac{1}{\cos ^2 x} d x=\int \sec ^2 x d x=\tan x+c$
Question 3.
(i) $12^3$
(ii) $\frac{x^{24}}{x^{25}}$
(iii) $\mathrm{e}^{\mathrm{x}}$
Solution:
(i) $\int 12^3 d x=12^3 \int d x=12^3 x+c$
(ii) $\int \frac{x^{24}}{x^{25}} d x=\int \frac{1}{x} d x=\log |x|+c$
(iii) $\int e^x d x=e^x+c$
Question 4.
(i) $\left(1+x^2\right)^{-1}$
(ii) $\left(1-x^2\right)^{-\frac{1}{2}}$

Solution:
(i) $\int\left(1+x^2\right)^{-1} d x=\int \frac{1}{1+x^2} d x=\tan ^{-1} x+c$
(ii) $\int\left(1-x^2\right)^{-1 / 2} d x=\int \frac{1}{\sqrt{1-x^2}} d x=\sin ^{-1} x+c$