Exercise · ZW-U-SUBSTITUTION-004 · ★★☆ Intermediate
Compute \(\displaystyle\int \frac{3 x^{2}}{x^{3} + 1}\,dx\).
Choose one option.
Show answer
- A. \(\ln{\left(\left|{x^{3} + 1}\right| \right)} + C\)
Explanation
Use the substitution \(u=x^{3}+1,\ du=3x^{2}\,dx\). The integral reduces to a basic form and returning to x gives \(\ln{\left(\left|{x^{3} + 1}\right| \right)} + C\). Check: \((\ln{\left(\left|{x^{3} + 1}\right| \right)})'=\frac{3 x^{2}}{x^{3} + 1}\).
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-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.
- ZW-U-SUBSTITUTION-009 If u=g(x), what is the differential du?