State the integration by parts formula.

For \(\int x e^x\,dx\), a useful choice is \(u=x\). What are \(du\), \(dv\), and \(v\)?

In the LIATE guideline, which is normally chosen first: algebraic or exponential?

For \(\int x\cos x\,dx\), choose \(u\) and \(dv\) using LIATE.

Evaluate \(\int x e^x\,dx\).

Evaluate \(\int x\cos x\,dx\).

Evaluate \(\int \ln x\,dx\) for \(x>0\).

Evaluate \(\int x\sin x\,dx\).

Evaluate \(\int x^2 e^x\,dx\).

Evaluate \(\int x^2\cos x\,dx\).

Evaluate \(\int_0^1 x e^x\,dx\).

Verify by differentiating that \(x\sin x+\cos x+C\) is an antiderivative of \(x\cos x\).

Evaluate \(\int x\ln x\,dx\) for \(x>0\).

Evaluate \(\int e^x\sin x\,dx\).

Find \(a\) if \(\int x e^{ax}\,dx=e^{ax}\left(\frac{x}{a}-\frac{1}{a^2}\right)+C\) and the integrand is \(xe^{3x}\).

For which polynomial degree is the tabular method especially efficient in \(\int p(x)e^x\,dx\), and why?

Evaluate \(\int x^3 e^x\,dx\) using repeated integration by parts or the tabular pattern.

A student evaluating \(\int x\cos x\,dx\) writes \(x\sin x-\int\sin x\,dx=x\sin x-\cos x+C\). Find and correct the sign error.

Explain why choosing \(u=e^x\) and \(dv=x\,dx\) is inefficient for \(\int xe^x\,dx\), even though it is not illegal.

Derive the integration by parts formula from the product rule.