use a change of variables or the table to evaluate the following definite integral\n int_{0}^{1}x^{2}e^{x^{3}…

use a change of variables or the table to evaluate the following definite integral\n int_{0}^{1}x^{2}e^{x^{3}}dx \nclick to view the table of general integration formulas.\n int_{0}^{1}x^{2}e^{x^{3}}dx=square \text{ (type an exact answer.) }

use a change of variables or the table to evaluate the following definite integral\n int_{0}^{1}x^{2}e^{x^{3}}dx \nclick to view the table of general integration formulas.\n int_{0}^{1}x^{2}e^{x^{3}}dx=square \text{ (type an exact answer.) }

Answer

Explanation:

Step1: Set substitution variable

Let $u = x^{3}$, then $du=3x^{2}dx$, and $x^{2}dx=\frac{1}{3}du$. When $x = 0$, $u=0^{3}=0$; when $x = 1$, $u = 1^{3}=1$.

Step2: Rewrite the integral

The integral $\int_{0}^{1}x^{2}e^{x^{3}}dx$ becomes $\frac{1}{3}\int_{0}^{1}e^{u}du$.

Step3: Integrate $e^{u}$

The antiderivative of $e^{u}$ is $e^{u}$. So, $\frac{1}{3}\int_{0}^{1}e^{u}du=\frac{1}{3}[e^{u}]_{0}^{1}$.

Step4: Evaluate the definite - integral

$\frac{1}{3}[e^{u}]_{0}^{1}=\frac{1}{3}(e^{1}-e^{0})=\frac{1}{3}(e - 1)$.

Answer:

$\frac{e - 1}{3}$