Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
Find $\int e^{x} \sin x d x$
✓
Answer
Take $e^x$ as the first function and $\sin x$ as second function.
Using Integrating by parts, we have
$\mathrm{I}=\int e^{x} \sin x d x=e^{x}(-\cos x)+\int e^{x} \cos x d x$
$= -e^x \cos x + I_1 ......(i)$
Taking $e^x$ and $\cos x$ as the first and second functions, respectively, in $I_1,$ we get
$I_{1}=e^{x} \sin x-\int e^{x} \sin x d x$
Substituting the value of $bI_1$ in $(i),$ we get
$I = -e^x \cos x + e^x \sin x -I$
$\Rightarrow~2I = e^x (\sin x - \cos x)$
Hence,$I=\int e^{x} \sin x d x=\frac{e^{x}}{2}(\sin x-\cos x)+C$
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