If $\smallint {e^x}\left( {1 + x} \right)\,\,.\,\,{\sec ^2}\,\,\left( {x{e^x}} \right)dx = f\left( x \right) + {\rm{constant}}$ then f(x) is equal to

${\rm{cos}}\,\,\left( {x{e^x}} \right)$

${\rm{sin}}\,\,\left( {x{e^x}} \right)$

$2\,\,{\tan ^{ - 1}}\,\,\left( x \right)$

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

1 Medium

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Application +4 | -1

\int {e^x}\left( {1 + x} \right)\,\,.\,\,{\sec ^2}\,\,\left( {x{e^x}} \right)dx = f\left( x \right) + \mathrm{constant}

then f(x) is equal to

View answer and explanation

Correct option: D

\tan \,\,\left( {x\;{e^x}} \right)

Option D (

\tan \,\left( {x\\;{e^x}} \right)

) is correct because by substituting

t = xe^x

, the integral simplifies and can be easily solved.