Question

Integrate with respect to $x$:
${\sec }^{3}x$

Answer 1

$I=\int {\sec }^{3}xdx$

$=\int \sec x.se{c}^{2}xdx$

using formula of integration by parts:

$I=\sec x.\int {\sec }^{2}dx-\int \left(\sec x{)}^1\right(\int {\sec }^{2}xdx)$

$=\sec x.tanx-\int \left(\sec x\tan x\right)\left(\tan x\right)dx$

$\sec x\tan x-\int \sec x.{\tan }^{2}xdx$

$\sec x\tan x-\int \sec x({\sec }^{2}x-1)dx$

$\sec x.\tan x-\int {\sec }^{3}xdx+\int \sec xdx$

$I=\sec x.\tan x-I+|n|\sec x+\tan x|+C$

$\Rightarrow 2I=\sec x.\tan x+|n|\sec x+\tan x|+C$

$\Rightarrow I=\frac{1}{2}\sec x.\tan +\frac{1}{2}|n|\sec x+\tan x|+C$

Identities used:

$\int f\left(x\right)g\left(x\right)dx=f\left(x\right)\int g\left(x\right)-\int f\left(x\right)\left[\int g\left(x\right)dx\right]dx$

$\int {\sec }^{2}xdx-\tan x+C$

$\int \sec xdx-In|\sec x+\tan x|+C$