Question

Integrate with respect to x:

a) ${\sin }^{2}x$

b) ${\cos }^{2}x$

c) $4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}$

d) $\frac{1}{\sqrt{x+3}+\sqrt{x+2}}$

Answer 1

a) $\int {\sin }^{2}xdx=\int \frac{(1-\cos 2x)}{2}dx$

$=\frac{x}{2}-\frac{\sin 2x}{4}+C$

b) $\int {\cos }^{2}xdx=\int \frac{1+\cos 2x}{2}dx$

$=\frac{x}{2}+\frac{\sin 2x}{4}+C$

c) $4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}=2\sin x\left(\cos \right(2x)+\cos (-x\left)\right)$ U\sing $2\cos A\cos B=\cos (A+B)+\cos (A-B)$

$=2\sin x\cos 2x+2\sin x\cos x=\sin 3x+\sin (-x)+\sin 2x=\sin 3x+\sin 2x-\sin x$

$\int 4\sin x\cos \frac{x}{2}\cos \frac{3x}{2}dx=\int (\sin 3x+\sin 2x-\sin x)dx$

$=-\frac{\cos 3x}{3}-\frac{\cos 2x}{2}+\cos x+C$

d) $\frac{1}{\sqrt{x+3}+\sqrt{x+2}}$

$=\frac{1}{\sqrt{x+3}+\sqrt{x+2}}\frac{\sqrt{x+3}-\sqrt{x+2}}{\sqrt{x+3}-\sqrt{x+2}}=\sqrt{x+3}-\sqrt{x+2}$

$\int (\sqrt{x+3}-\sqrt{x+2})dx=\frac{2}{3}\left(\right(x+3{)}^{\frac{3}{2}}-(x+2{)}^{\frac{3}{2}})+C$