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

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

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

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

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

Solution steps

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

Simplify

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

Apply u-substitution

: \int - u^{4} d u

= \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}

= - \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

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