Evalute : e^x (1 x -1 x^2 ) d x - Vidyadip

Question

Evalute : $\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right) d x$

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Answer

Let $I=\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right) d x$
Put $f(x)=\frac{1}{x}$. Then $f^{\prime}(x)=-\frac{1}{x^2}$
$
\begin{aligned}
\therefore I & =\int e^x-\left[f(x)+f^{\prime}(x)\right] d x \\
& =e^x \cdot f(x)+c= e ^x \cdot \frac{1}{x}+c .
\end{aligned}
$

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