Mathematics

Solve:-
$$\displaystyle\int {\dfrac{{{e^x}(1 + x)}}{{{{\cos }^2}(x{e^x})}}} dx$$


ANSWER

$$I=\tan \left( x{{e}^{x}} \right)+C$$


SOLUTION

Consider the given integral.


$$I=\displaystyle\int{\dfrac{{{e}^{x}}\left( 1+x \right)}{{{\cos }^{2}}\left( x{{e}^{x}} \right)}}dx$$


 


Let $$t=x{{e}^{x}}$$


$$ \dfrac{dt}{dx}=x{{e}^{x}}+{{e}^{x}} $$


$$ dt={{e}^{x}}\left( x+1 \right)dx $$


 


Therefore,


$$ I=\displaystyle\int{\dfrac{dt}{{{\cos }^{2}}t}} $$


$$ I=\displaystyle\int{{{\sec }^{2}}tdt} $$


$$ I=\tan t+C $$


 


On putting the value of $$t$$, we get


$$I=\tan \left( x{{e}^{x}} \right)+C$$


 


Hence, this is the answer.

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Single Correct Medium Published on 17th 09, 2020
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