Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 275: 24

$$\frac{1}{3}\sin (x^{3}+1) +c $$

Given $$\int x^{2} \cos \left(x^{3}+1\right) d x$$ Let $$ u= x^{3}+1\ \ \ \Rightarrow \ \ \ du=3x^2dx$$ Then \begin{align*} \int x^{2} \cos \left(x^{3}+1\right) d x&= \int \frac{1}{3} \cos \left(u\right)du\\ &= \frac{1}{3}\sin u +c\\ &=\frac{1}{3}\sin (x^{3}+1) +c \end{align*}