indefinite integral What is the indefinite integral of sin3

indefinite integral :

What is the indefinite integral of sin^3x?

Solution

Int (sin x)^3dx

We\'ll write (sin x)^3 = (sin x)^2*(sin x)

We\'ll re-write the integral:

Int (sin x)^3dx = Int (sin x)^2*(sin x)dx

From the fundamental formula of trigonometry, we\'ll get:

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

We\'ll re-write the integral:

Int [1 - (cos x)^2]*(sin x)dx

We\'ll substitute cos x = t.

We\'ll differentiate both sides:

-sin xdx = dt

We\'ll re-write the integral in t:

- Int (1 - t^2)dt = -Int dt + Int t^2

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