Exercise · ZW-INTEGRATION-BY-PARTS-005 · ★★★ Advanced
Compute \(\displaystyle\int x^{2} e^{x}\,dx\).
Choose one option.
Show answer
- C. \(\left(x^{2} - 2 x + 2\right) e^{x} + C\)
Explanation
Choose \(u=x^{2},\ dv=e^{x}dx\ (\text{twice})\) and apply \(\int u\,dv=uv-\int v\,du\). The result is \(\left(x^{2} - 2 x + 2\right) e^{x} + C\). Check: \((\left(x^{2} - 2 x + 2\right) e^{x})'=x^{2} e^{x}\).
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-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?