Mathematics

# Evaluate $\displaystyle \int \dfrac{e^x(1+x)}{\cos^2(xe^x)}dx$ on $I \subset R$ \ $\{x\in R:\cos(xe^x)=0\}$

##### SOLUTION
$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx}$
Let
$x \; {e}^{x} = u$
$\left( {e}^{x} + x \; {e}^{x} \right) dx= du$
$\Rightarrow {e}^{x} \left( 1 + x \right) dx = du$
Thus the integral will become-
$\int{\cfrac{du}{\cos^{2}{u}}}$
$= \int{\sec^{2}{u} \; du}$
$= \tan{u} + C$
Substituting the value of $u$ in above equation, we get
$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx} = \tan{\left( x \; {e}^{x} \right)} + C$
Hence
$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx} = \tan{\left( x \; {e}^{x} \right)} + C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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