Question

` \int \text{ x} \text{ sec x}^2 \text{  dx  is  equal  to }`

 

Options

• \[\frac{1}{2}\] log (sec x² + tan x²) + C

• \[\frac{x^2}{2}\]  log (sec x² + tan x²) + C

• 2 log (sec x² + tan x²) + C

• none of these

Answer 1

\[\frac{1}{2}\]  log (sec x² + tan x²) + C

\[\text{ Let I }= \int x \sec x^2 dx\]
\[\text{ Putting x}^2 = t\]
\[ \Rightarrow 2x \text{ dx }= dt\]
\[ \Rightarrow x \text{ dx} = \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\sec t \cdot dt\]
\[ = \frac{1}{2} \text{ log } \left| \sec t + \tan t \right| + C\]
\[ = \frac{1}{2} \text{ log }\left| \sec x^2 + \tan x^2 \right| + C \left( \because t = x^2 \right)\]