Exercise · ZW-INTEGRATION-BY-PARTS-001 · ★★☆ Intermediate
When computing \(\displaystyle\int x\,e^{x}\,dx\) by parts, which choice of u and dv is suitable?
Choose one option.
Show answer
- C. \(u=x,\ dv=e^{x}\,dx\)
Explanation
It is best to choose u as the part that simplifies when differentiated: \(u=x\) gives \(du=dx\), and \(dv=e^{x}\,dx\) integrates easily to \(v=e^{x}\). The other choice makes the integral harder.
Review the concept: Integration by parts
Integration by parts is a technique for integrating products of functions, derived from the product rule. It transforms \(\int u\,dv\) into \(uv-\int v\,du\).
\[\int u\,dv=uv-\int v\,du\]
More practice problems: integration by parts
- ZW-INTEGRATION-BY-PARTS-002 Compute ∫ x e^x dx.
- ZW-INTEGRATION-BY-PARTS-003 Compute ∫ x sin x dx.
- ZW-INTEGRATION-BY-PARTS-004 Compute ∫ ln (x ) dx.
- ZW-INTEGRATION-BY-PARTS-005 Compute ∫ x^2 e^x dx.
- ZW-INTEGRATION-BY-PARTS-006 Compute ∫ x cos x dx.
- ZW-INTEGRATION-BY-PARTS-007 What is the integration by parts formula?
- ZW-INTEGRATION-BY-PARTS-008 According to the LIATE rule, which family of functions should be chosen first as u?
- ZW-INTEGRATION-BY-PARTS-009 Which choice of u and dv is suitable for ∫ xln x dx?