Question

Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?

Options

• \[I=e^x\cos x-\int e^x\cos x\,dx\]

• \[I=-e^x\cos x+\int e^x\cos x\,dx\]

• \[I=-e^x\sin x-\int e^x\cos x\,dx\]

• \[I=e^x\sin x+\int e^x\cos x\,dx\]

Answer 1

Choose \(u=e^x\) and integrate \(\sin x\) to obtain \(-\cos x\). Therefore \(I=-e^x\cos x+\int e^x\cos x\,dx\).