What is the antiderivative of function y=2sinx*cos^3x?

The antiderivative of the given function is the primitive
function that could be determined calculating the indefinite integral of
y.

We'll solve this integral using substitution
technique.

Let sin x = t => cos dx =
dt

We'll use Pythagorean identity to write (cos x)^2, with
respect to (sin x)^2:

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

We'll get the indefinite
integral:

`int` 2sin x*(cos x)^2* cos x dx = 2 `int` sin
x*[1 - (sin x)^2]* cos x dx

`int` 2t*(1-t^2)dt = 2`int` t
dt - 2 `int` t^3 dt

2 `int` 2t*(1-t^2)dt = 2t^2/2 - 2t^4/4
+ C

`int` 2t*(1-t^2)dt = t^2 - t^4/2 +
C

`int` 2sin x*(cos x)^2* cos x dx = (sin x)^2 - (sin
x)^4/2 + C

The antiderivative of the given
function is the primitive function Y = (sin x)^2 - (sin x)^4/2 +
C.