Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Find $\int e^{x} \sin x d x$
✓
Answer
Take ex as the first function and sin x as second function. Using Integrating by parts, we have $\mathrm{I}=\int e^{x} \sin x d x=e^{x}(-\cos x)+\int e^{x} \cos x d x$ = -ex cos x + $I_1$ ......(i) Taking ex 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 $I_1$ in (i), we get $I$ = -ex cos x + ex sin x - $I$ $\Rightarrow~2I$ = ex (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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