Exercise · ZW-INTEGRATION-BY-PARTS-010 · ★☆☆ Basic
If \(u=x\) and \(dv=\cos x\,dx\), what are \(du\) and \(v\)?
Choose one option.
Show answer
- B. \(du=dx,\ v=\sin x\)
Explanation
Differentiating \(u=x\) gives \(du=dx\), and integrating \(\cos x\) gives \(v=\sin x\), since \(\int\cos x\,dx=\sin x\).
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?