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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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