Exercise · ZW-U-SUBSTITUTION-003 · ★☆☆ Basic
Compute \(\displaystyle\int x e^{x^{2}}\,dx\).
Choose one option.
Show answer
- B. \(\frac{e^{x^{2}}}{2} + C\)
Explanation
Use the substitution \(u=x^{2},\ du=2x\,dx\). The integral reduces to a basic form and returning to x gives \(\frac{e^{x^{2}}}{2} + C\). Check: \((\frac{e^{x^{2}}}{2})'=x e^{x^{2}}\).
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-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.
- ZW-U-SUBSTITUTION-009 If u=g(x), what is the differential du?