QuestionIf we perform the appropriate (i.e. helpful) $u$-sub for $\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}$, what does the new integral look like in terms of $u$ right after performing the substitution? $\int x^{2} \sin \left(u^{3}\right) d u$ $\int \frac{1}{3} \sin (u) d u$ none of these $\int u^{2} \sin \left(u^{3}\right) d u$

Studdy Solution
Substitute $u = x^3$ and $dx = \frac{du}{3x^2}$ into the original integral:
$$\int x^2 \sin(x^3) \, dx = \int x^2 \sin(u) \cdot \frac{du}{3x^2}$$
The $x^2$ terms cancel out, simplifying to:
$$\int \frac{1}{3} \sin(u) \, du$$
The new integral in terms of $u$ is:
$$\int \frac{1}{3} \sin(u) \, du$$
The correct choice is:
$\int \frac{1}{3} \sin(u) \, du$