What's the integral of cos^3(x)?

The integral of cos^3(x) is (sin(x) + sin(3x))/4 + C.

To find the integral of cos^3(x), we can use the trigonometric identity cos^3(x) = cos(x)cos^2(x). We can then use integration by substitution, letting u = sin(x) and du = cos(x)dx. This gives us:

∫cos^3(x)dx = ∫cos(x)cos^2(x)dx
Let u = sin(x), du = cos(x)dx
= ∫(1-u^2)du
= u - (1/3)u^3 + C
= sin(x) - (1/3)sin^3(x) + C

However, we can simplify this further using the identity sin^2(x) = 1 - cos^2(x). This gives us:

sin^3(x) = sin(x)sin^2(x)
= sin(x)(1 - cos^2(x))
= sin(x) - sin(x)cos^2(x)

Substituting this back into our integral, we get:

∫cos^3(x)dx = sin(x) - (1/3)(sin(x) - sin(x)cos^2(x)) + C
= sin(x) - (1/3)sin(x) + (1/3)sin(x)cos^2(x) + C
= (sin(x) + sin(3x))/4 + C

Therefore, the integral of cos^3(x) is (sin(x) + sin(3x))/4 + C.