Example 8. Find $\int x^{2} \sin \left(x^{3}+1\right) d x$.
Solution. Let's make the substitution $x^{3}+1=u$. Differentiating this equality, we get $3 x^{2} d x=d u$. Hence, $x^{2} d x=\frac{1}{3} d u$. The integral takes a tabular form:

$$
\int x^{2} \sin \left(x^{3}+1\right) d x=\frac{1}{3} \int \sin u d u=-\frac{1}{3} \cos u+C=-\frac{1}{3} \cos \left(x^{3}+1\right)+C .
$$

The substitution $x^{3}+1=u$ is equivalent to bringing $x^{3}+1$ under the differential sign.