Integration of coses³x

The integration of cos^3(x) can be solved using a trigonometric identity. Here's the integration step by step: ∫cos^3(x) dx = ∫(cos^2(x) * cos(x)) dx Now, you can use the identity cos^2(x) = 1 - sin^2(x): = ∫(1 - sin^2(x)) * cos(x) dx = ∫cos(x) dx - ∫(sin^2(x) * cos(x)) dx The first integral, ∫cos(x) dx, is straightforward and equals sin(x): = sin(x) - ∫(sin^2(x) * cos(x)) dx Now, for the remaining integral, you can use a u-substitution. Let u = sin(x), so du = cos(x) dx: = sin(x) - ∫(u^2 * du) Now, integrate u^2 with respect to u: = sin(x) - ∫u^2 du = sin(x) - (u^3 / 3) + C Now, replace u with sin(x): = sin(x) - (sin^3(x) / 3) + C So, the integral of cos^3(x) is: ∫cos^3(x) dx = sin(x) - (sin^3(x) / 3) + C, where C is the constant of integration.