Exercise · ZW-U-SUBSTITUTION-006 · ★★☆ Intermediate
Compute \(\displaystyle\int \left(2 x + 1\right) \left(x^{2} + x\right)^{4}\,dx\).
Choose one option.
Show answer
- C. \(\frac{\left(x^{2} + x\right)^{5}}{5} + C\)
Explanation
Use the substitution \(u=x^{2}+x,\ du=(2x+1)\,dx\). The integral reduces to a basic form and returning to x gives \(\frac{\left(x^{2} + x\right)^{5}}{5} + C\). Check: \((\frac{\left(x^{2} + x\right)^{5}}{5})'=x^{4} \left(x + 1\right)^{4} \left(2 x + 1\right)\).
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?
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- 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-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?