Question

Evaluate \[\int e^x\sin x\,dx.\]

Options

• \[\frac{e^x}{2}(\sin x-\cos x)+C\]

• \[\frac{e^x}{2}(\sin x+\cos x)+C\]

• \[\frac{e^x}{2}(\cos x-\sin x)+C\]

• \[e^x(\sin x-\cos x)+C\]

Answer 1

A second application gives \(\int e^x\cos x\,dx=e^x\sin x-I\). Hence \(2I=e^x(\sin x-\cos x)\), so the required integral is \(\frac{e^x}{2}(\sin x-\cos x)+C\).