Exercise · ZW-U-SUBSTITUTION-008 · ★★★ Advanced
Compute \(\displaystyle\int_{0}^{1} \frac{x}{1+x^{2}}\,dx\).
Choose one option.
Show answer
- A. \(\frac{\ln 2}{2}\)
Explanation
With \(u=1+x^{2}\), \(du=2x\,dx\) and limits from \(1\) to \(2\), the integral becomes \(\frac{1}{2}\int_{1}^{2}\frac{du}{u}=\frac{\ln 2}{2}\). Changing the limits avoids going back to \(x\).
Review the concept: Integration by substitution
Technique that simplifies an integral through the change of variable \(u=g(x)\), \(du=g^{\prime}(x)\,dx\).
\[\int f(g(x))\,g'(x)\,dx=\int f(u)\,du\]
More practice problems: integration by substitution
- ZW-U-SUBSTITUTION-001 When is it convenient to use integration by substitution?
- ZW-U-SUBSTITUTION-002 Compute ∫ 2 x cos x^2 dx.
- ZW-U-SUBSTITUTION-003 Compute ∫ x e^x^2 dx.
- ZW-U-SUBSTITUTION-004 Compute ∫ (3 x^2)/(x^3 + 1) dx.
- ZW-U-SUBSTITUTION-005 Compute ∫ sin x cos ^2 x dx.
- ZW-U-SUBSTITUTION-006 Compute ∫ (2 x + 1) (x^2 + x)^4 dx.
- ZW-U-SUBSTITUTION-007 Compute ∫ e^3 x dx.
- ZW-U-SUBSTITUTION-009 If u=g(x), what is the differential du?