Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 18

$cos(cos(x))+C$

Evaluate the Integral using substitution: $\int sinxsin(cosx)dx$ Substitution Rule: $\int f(g(x))g’(x)dx = \int f(u)du$ $u= cosx$ $du =-sinx$ Since $du$ in the expression is $sinx$ it must be multiplied by $-1$ Solve the integral in terms of $u$: $\int sin(u)(-1)du$ $-1\int sin(u)du $ $-1(-cos(u)) +C$ $cos(u) + C$ Substitute for $u$: $cos(cos(x)) +C$