What is the antiderivative of the function f(x) = sin^3 x?

We'll calculate the antiderivative F(x) integrating the
given function.

F(x) is a function such as dF/dx =
f(x).

Int f(x)dx = F(x) +
C

We'll write the function as a
product:

(sinx)^3 = (sinx)^2*sin
x

We'll integrate both
sides:

Int (sinx)^3dx = Int [(sinx)^2*sin
x]dx

We'll write (sinx)^2  = 1 -
(cosx)^2

Int [(sinx)^2*sin x]dx = Int [(1 - (cosx)^2)*sin
x]dx

We'll remove the
brackets:

Int [(1 - (cosx)^2)*sin x]dx  = Int sin xdx - Int
(cosx)^2*sin xdx

We'll solve Int (cosx)^2*sin xdx using
substitution technique:

cos x =
t

We'll differentiate both
sides:

cos xdx = dt

We'll
re-write the integral, changing the variable:

Int
(cosx)^2*sin xdx = Int t^2dt

Int t^2dt = t^3/3 +
C

Int (cosx)^2*sin xdx = (cos x)^3/3 +
C

Int (sinx)^3dx = Int sin xdx - Int (cosx)^2*sin
xdx

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