What is the integral of sin(x)(cos(x))^3 ?

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

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integral of sin(x)(cos(x))^3

\int sin (x) (cos (x))^{3} d x

- \frac{cos ^{4} ( x )}{4} + C

Solution steps

\int sin (x) (cos (x))^{3} d x

Apply u-substitution

: \int - u^{3} d u

= \int - u^{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= - \frac{u ^{4}}{4}

Substitute back

u = cos (x)

= - \frac{cos ^{4} ( x )}{4}

Add a constant to the solution

= - \frac{cos ^{4} ( x )}{4} + C

f (x) = \frac{1}{4 x}

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What is the integral of sin(x)(cos(x))^3 ?
The integral of sin(x)(cos(x))^3 is -(cos^4(x))/4+C