Question

Evaluate:

${\displaystyle \int e^x\sin x dx}$

Answer 1

Let I = ${\displaystyle \int e^x\sin x dx}$   ....(1)

Integrating by parts,

I = ${\displaystyle e^x\int \sin x dx-\int (\int \sin x dx\frac{d}{dx}{e}^x)dx}$

= ${\displaystyle e^x(-\cos x)-\int -\cos x\cdot e^{x}dx}$

= ${\displaystyle -e^x\cos x+\int e^x\cos x dx}$

Again integrating by parts,

I = ${\displaystyle -e^x\cos x+e^x\int \cos x dx-\int (\int \cos x dx\frac{d}{dx}{e}^x)dx}$

I = ${\displaystyle -e^x\cos x+e^x\sin x-\int \sin x{e}^{x}dx}$

∴ I = ${\displaystyle -e^x\cos x+e^x\sin x-I}$  ....[From (1)]

∴ 2I = ${\displaystyle -e^x(\cos x-\sin x)+c_1}$

∴ I = ${\displaystyle \frac{e^x}{2}(\sin x-\cos x)+c (\text{where} c=\frac{c_1}{2})}$