Question

Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?

Options

• \[\int f(x)g(x)\,dx=f(x)g(x)-\int f'(x)g'(x)\,dx\]

• \[\int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int\left[f'(x)\int g(x)\,dx\right]dx\]

• \[\int f(x)g(x)\,dx=f'(x)\int g(x)\,dx+\int\left[f(x)\int g(x)\,dx\right]dx\]

• \[\int f(x)g(x)\,dx=g(x)\int f(x)\,dx-\int\left[g'(x)\int f(x)\,dx\right]dx\]

Answer 1

Take \(u=f(x)\) and \(dv=g(x)\,dx\) in \(\int u\,dv=uv-\int v\,du\). Thus \(du=f'(x)\,dx\) and \(v=\int g(x)\,dx\), giving the stated expression.