How do you find the antiderivative of ((cosx)^2)*(sinx)?

1 Answer
Jul 13, 2016

int cos^2(x) sin(x) dx = -(cos^3(x))/(3) + C

Explanation:

In order to evaluate this integral, we'd have to perform a u-substitution.

Let's write out our integral first, before we proceed:

int cos^2(x) sin(x) dx

When performing a u-substitution, the goal in mind is to find a factor which is the derivative of, namely du. In this case, we can let

u = cos(x), so then du = -sin(x) dx, or simply -du = sin(x) dx

The reason why we've chosen u = cos(x) and not u = cos^2(x) is because when integrating, we can simply integrate u itself, whatever power it has, giving us

int u^2 * -du = -int u^2 du = -(u^3)/(3)+C = -(cos^3(x))/(3) + C

Since integration and derivation are related by the Fundamental Theorem of Calculus, we can even check our answer by taking a derivative.

Checking our answer:

d/dx[-1/3 * cos^3(x)] =-cos^2(x) * -sin(x) = cos^2(x) sin(x)