Exercise · ZW-U-SUBSTITUTION-010 · ★★☆ Intermediate
Compute \(\displaystyle\int \frac{\ln x}{x}\,dx\).
Choose one option.
Show answer
- C. \(\frac{(\ln x)^{2}}{2} + C\)
Explanation
With \(u=\ln x\), \(du=\frac{dx}{x}\), the integral becomes \(\int u\,du=\frac{u^{2}}{2}+C\). Returning to \(x\) gives \(\frac{(\ln x)^{2}}{2}+C\). Check: \(\left(\frac{(\ln x)^{2}}{2}\right)'=\frac{\ln x}{x}\).
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-008 Compute ∫_0^1 (x)/(1+x^2) dx.