Exercise 13

Consider the indefinite integral $\int \sin^3(x) \, dx$.

1. Explain why the substitution $u = \sin(x)$ will not work to help evaluate the given integral.

2. Recall the Fundamental Trigonometric Identity, which states that $\sin^2(x) + \cos^2(x) = 1$. By observing that $\sin^3(x) = \sin(x) \cdot \sin^2(x)$, use the Fundamental Trigonometric Identity to rewrite the integrand as the product of $\sin(x)$ with another function.

3. Explain why the substitution $u = \cos(x)$ now provides a possible way to evaluate the integral in (b).

4. Use your work in (a)-(c) to evaluate the indefinite integral $\int \sin^3(x) \, dx$.

5. Use a similar approach to evaluate $\int \cos^3(x) \, dx$.