Evaluate the definite integral integral01 x2 (x3 + 2)3 \,dx using substitution.

Let $u = x^3 + 2$.

Differentiate $u$ with respect to $x$:

Rearrange to find $dx$ in terms of $du$:

Since the integral is definite, we need to change the limits from $x$-values to $u$-values.

• When $x = 0$: $$u = (0)^3 + 2 = 0 + 2 = 2$$
• When $x = 1$: $$u = (1)^3 + 2 = 1 + 2 = 3$$

So the new limits of integration are from $u=2$ to $u=3$.

Substitute $u$ and $x^2 \,dx$ into the integral:

Apply the power rule for integration, $\int u^n \,du = \frac{u^{n+1}}{n+1}$:

Substitute the upper limit and subtract the result of substituting the lower limit:

The final answer is $\frac{65}{12}$