Q. If $\int_{}^{}e^{x}(1 + x).\sec^{2}{(xe^{x})}dx = f(x) + constant$, then $f(x)$ is equal to
A) $cos(xe^{x})$
B) $sin(xe^{x})$
C) $2\tan^{- 1}{(x)}$
D) $\tan{(x\ e^{x})}$
Explanation
(d)
Given, $\int_{}^{}e^{x}(1 + x).\sec^{2}{(xe^{x})}dx = f(x) + constant$
Put $xe^{x} = t \Rightarrow \left( e^{x} + xe^{x} \right)dx = dt$ in LHS, we get
LHS = $\int_{}^{}{\sec^{2}t}\ dt$
$= \tan t +$ constant
$\Rightarrow \tan{(xe^{x})} + constant = f(x) + constant$
$\Rightarrow f(x) = \tan{(xe^{x})}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.