Consider the indefinite integral ∫sin^3(x) dx

Question

a) Explain why the substitution u = sin (x) will not work to help evaluate the integral.

b) By observing that sin³(x) = sin(x) • sin²(x), and using the Pythagorean Identity, rewrite the integral as a product of sin(x) and another function.

c) Now explain why the substitution u = cos(x) now helps evaluate the integral.

d) Use your work in parts a), b), and c) to evaluate the integral, thereby discovering the antiderivative of sin³(x)

e) Use a similar approach to evaluate ∫ cos³(x) dx.

Answer 1

a) Because you will need a cos as part of du.

b) sin x * sin²x = sin x(1- cos²)

and now you can integrate sin x by itself and the the substitution of u=cos x , du=-sin x dx makes your integrand -u du which is easy to integrate.