Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 86

$$\frac{(f(x))^{4}}{4}+C$$

Given $$\int f(x)^{3} f^{\prime}(x) \mathrm{d} x$$ Let $$ u=f(x) \ \ \ \Rightarrow \ \ \ du = f'(x) dx$$ Then \begin{aligned} \int f(x)^{3} f^{\prime}(x) \mathrm{d} x &=\int u^{3} \mathrm{d} u \\ &=\frac{u^{4}}{4}+C \\ &=\frac{(f(x))^{4}}{4}+C \end{aligned}