Question

find the indefinite integral.

$$ int x ^ { 2 } e ^ { x ^ { 3 } } d x $$

$$ int x ^ { 2 } e ^ { x ^ { 3 } } d x = square $$

Explanation:

Step1: Use substitution

Let \(u = x^{3}\), then \(du=3x^{2}dx\), and \(x^{2}dx=\frac{1}{3}du\).

Step2: Substitute into the integral

\(\int x^{2}e^{x^{3}}dx=\int e^{u}\cdot\frac{1}{3}du\).

Step3: Integrate

\(\int e^{u}\cdot\frac{1}{3}du=\frac{1}{3}\int e^{u}du\). Since \(\int e^{u}du = e^{u}+C\), we have \(\frac{1}{3}e^{u}+C\).

Step4: Back - substitute

Substitute \(u = x^{3}\) back, we get \(\frac{1}{3}e^{x^{3}}+C\).

Answer:

\(\frac{1}{3}e^{x^{3}}+C\)