Question

Evaluate: \( \int \sec^3 x \, dx \)

Hint:

For \( \int \sec^3 x \, dx \), use integration by parts and trigonometric identities; standard result involves \( \sec x \tan x \) and \( \ln |\sec x + \tan x| \).

Answer 1

Use integration by parts. Let: 
\[ u = \sec x, dv = \sec^2 x \, dx \Rightarrow du = \sec x \tan x \, dx, v = \tan x. \] \[ \int \sec^3 x \, dx = \sec x \tan x - \int \tan x \cdot \sec x \tan x \, dx = \sec x \tan x - \int \sec x \tan^2 x \, dx. \] Rewrite: \( \tan^2 x = \sec^2 x - 1 \). 
\[ \int \sec x \tan^2 x \, dx = \int \sec x (\sec^2 x - 1) \, dx = \int \sec^3 x \, dx - \int \sec x \, dx. \] \[ \int \sec^3 x \, dx = \sec x \tan x - \left( \int \sec^3 x \, dx - \int \sec x \, dx \right). \] \[ 2 \int \sec^3 x \, dx = \sec x \tan x + \int \sec x \, dx. \] \[ \int \sec x \, dx = \ln |\sec x + \tan x| + c. \] \[ \int \sec^3 x \, dx = \frac{1}{2} \left( \sec x \tan x + \ln |\sec x + \tan x| \right) + c. \] Answer: \( \frac{1}{2} \sec x \tan x + \frac{1}{2} \ln |\sec x + \tan x| + c \).