Indefinite Integration — Maths STD 12 Science — Question

$\begin{aligned} & I=\int \sec ^3 x d x \\ & I=\int \sec x \cdot \sec ^2 x d x \\ & I=\sec x \cdot \int \sec ^2 x d x-\int\left[\frac{d}{d x}(\sec x) \cdot \int \sec ^2 x d x\right] d x \\ & I=\sec x \cdot \tan x-\int \sec x \cdot \tan x \cdot \tan x d x \\ & I=\sec x \cdot \tan x-\int \sec x\left(\sec ^2 x-1\right) d x \\ & I=\sec x \cdot \tan x-\int\left[\sec ^3 x-\sec x\right] d x \\ & I=\sec x \cdot \tan x-\int \sec ^3 x+\int \sec x d x \\ & I=\sec x \cdot \tan x-I+\log |\sec x+\tan x|+c \\ & 2 I=\sec x \cdot \tan x+\log |\sec x+\tan x|+c \\ & \therefore I=\frac{1}{2}(\sec x \cdot \tan x+\log |\sec x+\tan x|)+c\end{aligned}$