Question

$$\int {\tan }^{3/2}x{\sec }^2\text{x dx}$$

Answer 1

$$\int {\tan }^{\frac{3}{2}}x\cdot {\sec }^2\text{x dx}$$
$$Let\tan x=t$$
$$\Rightarrow {\sec }^{2}x=\frac{dt}{dx}$$
$$\Rightarrow {\sec }^2\text{x dx}=dt$$
$$Now,\int {\tan }^{\frac{3}{2}}x\cdot {\sec }^2\text{x dx}$$
$$=\int t^{\frac{3}{2}}dt$$

${\displaystyle =\left[\frac{t^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right] +C}$

$$=\frac{2}{5}{t}^{\frac{5}{2}}+C$$
$$=\frac{2}{5}{\tan }^{\frac{5}{2}}x+C$$