Exercise · ZW-INTEGRATION-BY-PARTS-009 · ★☆☆ Basic
Which choice of \(u\) and \(dv\) is suitable for \(\displaystyle\int x\ln x\,dx\)?
Choose one option.
Show answer
- C. \(u=\ln x,\ dv=x\,dx\)
Explanation
Differentiating \(\ln x\) gives \(1/x\), which simplifies the integral. Also \(dv=x\,dx\) integrates easily to \(v=x^2/2\).
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-001 When computing ∫ x e^x dx by parts, which choice of u and dv is suitable?
- 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?