What is the integral of sin(x)(cos(x))^4 ?

The integral of sin(x)(cos(x))^4 is -(cos^5(x))/5+C

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integral of sin(x)(cos(x))^4

\int sin (x) (cos (x))^{4} d x

- \frac{cos ^{5} ( x )}{5} + C

Solution steps

\int sin (x) (cos (x))^{4} d x

Simplify

= \int cos^{4} (x) sin (x) d x

Apply u-substitution

= \int - u^{4} d u

Take the constant out:

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

= - \int u^{4} d u

Apply the Power Rule

= - \frac{u ^{5}}{5}

Substitute back

u = cos (x)

= - \frac{cos ^{5} ( x )}{5}

Add a constant to the solution

= - \frac{cos ^{5} ( x )}{5} + C

What is the integral of sin(x)(cos(x))^4 ?
The integral of sin(x)(cos(x))^4 is -(cos^5(x))/5+C