Mathematics

# Evaluate the following integral:$\displaystyle \int { \cfrac { \left( x+1 \right) { e }^{ x } }{ \cos ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx\quad$

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

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

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

$=\displaystyle\int{{\sec}^{2}{t}\,dt}$

$=\tan{t}+c$

$=\tan{\left(x{e}^{x}\right)}+c$ where $t=x{e}^{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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The average value of a function f(x) over the interval, [a,b] is the number $\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$
The square root $\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$ is called the root mean square of f on [a, b]. The average value of $\displaystyle \mu$ is attained id f is continuous on [a, b].