What is the integral of (cos(x))^5sin(x) ?

The integral of (cos(x))^5sin(x) is -(cos^6(x))/6+C

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integral of (cos(x))^5sin(x)

\int (cos (x))^{5} sin (x) d x

- \frac{cos ^{6} ( x )}{6} + C

Solution steps

\int (cos (x))^{5} sin (x) d x

Simplify

= \int cos^{5} (x) sin (x) d x

Apply u-substitution

= \int - u^{5} d u

Take the constant out:

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

= - \int u^{5} d u

Apply the Power Rule

= - \frac{u ^{6}}{6}

Substitute back

u = cos (x)

= - \frac{cos ^{6} ( x )}{6}

Add a constant to the solution

= - \frac{cos ^{6} ( x )}{6} + C

What is the integral of (cos(x))^5sin(x) ?
The integral of (cos(x))^5sin(x) is -(cos^6(x))/6+C