Question

Find $\int \frac{1}{sinxco{s}^{3}x}dx$

Answer 1

Let $I=\int \frac{1}{\sin x{\cos }^{3}x}dx$

$⟹I=\int \frac{1}{\frac{\sin x}{\cos x}{\cos }^{4}x}dx$

$⟹I=\int \frac{{\sec }^{2}x{\sec }^{2}xdx}{\tan x}$

Put $\tan x=t⟹{\sec }^{2}xdx=dt$, we get

$I=\int \frac{dt(t^2+1)}{t}$ .......... $(\because {\sec }^{2}x-{\tan }^{2}x=1)$

$⟹I=\int tdt+\frac{dt}{t}$

$=\frac{t^2}{2}+\ln t+c$

$=\frac{{\tan }^{2}x}{2}+\ln |\tan x|+c$

Hence, $=\int \frac{1}{\sin x{\cos }^{3}x}dx=\frac{{\tan }^{2}x}{2}+\ln |\tan x|+c$