Chapter 7 - Techniques of Integration - 7.2 Trigonometric Integrals - 7.2 Exercises - Page 524: 16

$\frac{\sin^3 x}{3}+C$

$\int \tan^2 x\cos^3 x\ dx$ $=\int\frac{\sin^2 x}{\cos^2 x}*\cos^3 x\ dx$ $=\int\sin^2 x\cos x\ dx$ Let $u=\sin x$. Then $du=\cos x\ dx$. $=\int u^2 du$ $=\frac{u^3}{3}+C$ $=\boxed{\frac{\sin^3 x}{3}+C}$