How to integrate $sec^{3}x$

Question

How do I integrate $sec^{3}x$. I have done this by converting $sec$ to $cos$ and then using $cos^{3}x= (1-sin^{2}x)cosx$ and then putting sinx=t and then using partial fractions. But this is very long method. I want shorter method for this

Thanks

Answer 1

Hint: Try integration by parts.

$$\int \sec^3 x \, dx=\int \sec x\, d \tan x$$

Answer 2

$$\int \sec^3 (x)\,dx$$

Hint:

Use the reduction formula, $\color{gray}{\displaystyle\int \sec^k(x)\,dx=\frac{\sin(x)\sec^{k-1}(x)}{k-1}+\frac{k-2}{k-1}\displaystyle\int \sec^{-2+k}(x)\,dx}$

$$=\frac 1 2 \tan(x)\sec(x)+\frac1 2 \int \sec(x)\,dx$$

Multiply numerator and denominator of $\sec(x)$ by $\tan(x)\sec(c)$:

$$=\frac 1 2 \tan(x)\sec(x)+\frac 1 2 \displaystyle\int \frac{\sec^2(x)+\sec(x)\tan(x)}{\sec(x)+\tan(x)}\,dx$$