Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
Integrate the function $\frac{x e^{x}}{(1+x)^{2}}$
✓
Answer
$I=\int \frac{x e^{x}}{(1+x)^{2}} d x=\int e^{x}\left\{\frac{x}{(1+x)^{2}}\right\} d x$ = $\int e^{x}\left\{\frac{1+x-1}{(1+x)^{2}}\right\} d x$ = $\int e^{x}\left\{\frac{1}{1+x}-\frac{1}{(1+x)^{2}}\right\} d x$ Now, Let $\frac{1}{1+x}=f(x) \Rightarrow f^{\prime}(x)=-\frac{1}{(1+x)^{2}}$ We know that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+C$ Thus, $\int \frac{x e^{x}}{(1+x)^{2}} d x=\frac{e^{x}}{1+x}+C$
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