Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 275: 23

$$\frac{1}{4}\sin (x^4) +c $$

Given $$\int x^{3} \cos \left(x^{4}\right) d x$$ Let $$ u= x^4\ \ \ \Rightarrow \ \ \ du=4x^3dx$$ Then \begin{align*} \int x^{3} \cos \left(x^{4}\right) d x&= \int \frac{1}{4} \cos \left(u\right) du\\ &= \frac{1}{4}\sin u +c\\ &=\frac{1}{4}\sin (x^4) +c \end{align*}