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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
MCQ
Solution
\[\frac{1}{2}\] log (sec x2 + tan x2) + C
\[\text{ Let I }= \int x \sec x^2 dx\]
\[\text{ Putting x}^2 = t\]
\[ \Rightarrow 2x \text{ dx }= dt\]
\[ \Rightarrow x \text{ dx} = \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\sec t \cdot dt\]
\[ = \frac{1}{2} \text{ log } \left| \sec t + \tan t \right| + C\]
\[ = \frac{1}{2} \text{ log }\left| \sec x^2 + \tan x^2 \right| + C \left( \because t = x^2 \right)\]
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