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
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