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[mathjax] Question 1 $=\int_{0}^{\pi /2} \frac{d x}{1+\sin x}$ Sol : $=\int_{0}^{\pi /2} \frac{d x}{1+\sin x}\times \frac{1-\sin x}{1-\sin x}$ $=\int_{0}^{\pi/2} \frac{(1-\sin x)dx}{1-\sin^2 x}$...
[mathjax] Question 1 $\int \frac{\cos xdx}{1+9\sin^2x}$ Sol : Given : $I=\int \frac{\cos xdx}{1+9\sin^2 x}$ $=\int \frac{\cos xdx}{(1)^2+(3\sin x )^2}$ 3sinx=z then 3cosxdx=dz $\cos...
[mathjax] Question 1 (i) $\int \frac{dx}{x^2-9}$ Sol : $I=\int \frac{dx}{x^2-9}$ Let $\frac{1}{(x-3)(x+3)}=\frac{A}{x-3}+\frac{B}{x+3}$ $\frac{1}{(x-3)(x+3)}=\frac{A(x+3)+B(x-3)}{(x-3)(x+3)}$...
[mathjax] Question 1 (i) $\int \frac{dx}{\sqrt{1+4x^2}}$ Sol : Given: $I=\int \frac{dx}{\sqrt{1+4x^2}}=\int \frac{dx}{\sqrt{(1)^2+(2x)^2}}$ $=\frac{dx}{\sqrt{1+(2x)^2}}$ $\left[\int...
[mathjax] Question 1 $\int e^{x}\left(\frac{1+x \log x}{x}\right) d x$ Sol : $I=\int e^{x}\left(\frac{1+x \log x}{x}\right) d x=\int e^{x}\left(\frac{1}{x}+\frac{x \log x}{x}\right) d x$ $I=\int...
[mathjax] Question 16 $\int x \sin x \cdot \sin 3 x d x$ Sol : $I=\int x \sin x \cdot \sin 3 x d x$ $I=\frac{1}{2}\left[x \frac{\sin 2 x}{2}-\int 1 \times \frac{\sin 2 x}{2} d...
[mathjax] $=\frac{1}{2}\left[\sin ^{2} x+\frac{1}{2} \cdot 2 \sin \theta \cdot \cos \theta\right]+c$ Now , $\int \frac{d x}{\sqrt{1-x^{2}}}=\int \frac{\cos \theta d \theta}{\sqrt{1-\sin ^{2}...
[mathjax] Question 1 $\int \frac{\cos x}{(1+\sin x)^{2}} d x$ Sol : Let z=1+sinx then dz=cosxdx Now , $\int \frac{\cos x}{(1+\sin^2 x)^{2}} d x$ $=\int \frac{d z}{z^{2}}$ $=\int z^{-2} d z$...
[mathjax] Question 1 $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$ Sol : Let z=√x then $d z=\frac{1}{2 \sqrt{x}} d x$ $2 d z=\frac{1}{\sqrt{x}} d x$ Question 2 $\int \frac{1}{x^{2}} \cdot \sin...
[mathjax] Question 1 $\int \sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}} d x$ Sol : $=\int \sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}} d x=\int \frac{\sin x}{\cos x} d x$ $=\int \tan x d x=\log |\sec x|+c$...
[mathjax] Question 6 $\int \sec ^{3} x \cdot \tan x d x$ Sol : Let z=secx then dz=secx.tanxdx Now , $\int \sec ^{3} x \cdot \tan x d x$ $=\int \sec ^{2} x \cdot \sec x \cdot \tan x d x$ $=\int z^{2}...
[mathjax] Question 1 $\displaystyle\int sin^{3}xdx$ Sol : Let z=cosx then dz=-sinxdx =$\displaystyle\int sin^{3}xdx$ =$\displaystyle\int sin^{2}x.sinxdx$ =$\displaystyle\int (1-cos^{2}x).sinxdx$...