x x ^3 x , , ,dx equal to - Vidyadip

MCQ

$\int {x\sin x\ {{\sec }^3}\ x\,\,\,dx} $ equal to

A
$\frac{1}{2}\left( {{{\sec }^2}\ x - \tan x} \right) + C$
✓
$\frac{1}{2}\left( {x\ {{\sec }^2}\ x - \tan x} \right) + C$
C
$\frac{1}{2}\left( {x\ {{\sec }^2}\ x + \tan x} \right) + C$
D
$\frac{1}{2}\left( {{{\sec }^2}\ x + \tan x} \right) + C$

✓

Answer

Correct option: B.

$\frac{1}{2}\left( {x\ {{\sec }^2}\ x - \tan x} \right) + C$

b
$\int x \sin x \sec ^{3} x d x=\int x \tan x \sec ^{2} x d x$

${=\left(x \frac{\tan ^{2} x}{2}\right)-\int\left(1 \cdot \frac{\tan ^{2} x}{2}\right) d x} $

${=\frac{x \tan ^{2} x}{2}-\frac{1}{2} \int\left(\sec ^{2} x-1\right) d x} $

${=\frac{x}{2} \tan ^{2} x-\frac{1}{2} \tan x+\frac{1}{2} x+c} $

${=\frac{x}{2} \sec ^{2} x-\frac{1}{2} \tan x+c} $

${=\frac{1}{2}\left(x \sec ^{2} x-\tan x\right)+c} $

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