Mathematics

# Write a value of $\int { { e }^{ ax }\left[ af(x)+f'(x) \right] } dx$

##### SOLUTION
$I=\displaystyle\int{{e}^{ax}\left[af\left(x\right)+{f}^{\prime}{\left(x\right)}\right]dx}$

$I=a\displaystyle\int{{e}^{ax}f\left(x\right)dx}+\displaystyle\int{{e}^{ax}{f}^{\prime}{\left(x\right)}dx}$

Let $u={e}^{ax}\Rightarrow\,du=a{e}^{ax}dx$

$dv={f}^{\prime}{\left(x\right)}\Rightarrow\,v=f\left(x\right)$

$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ]$......by parts formula.

$I=a\displaystyle\int{{e}^{ax}f\left(x\right)dx}+{e}^{ax}f\left(x\right)-a\displaystyle\int{{e}^{ax}f\left(x\right)dx}+c$

$\therefore\,I={e}^{ax}f\left(x\right)+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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