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

The integral of cos^5(x)(-sin(x)) is 1/6 cos^6(x)+C

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

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

\frac{1}{6} cos^{6} (x) + C

Solution steps

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

Simplify

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

Take the constant out:

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

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

Apply u-substitution

: \int - u^{5} d u

= - \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}

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

Substitute back

u = cos (x)

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

Simplify

- (- \frac{cos ^{6} ( x )}{6}) : \frac{1}{6} cos^{6} (x)

= \frac{1}{6} cos^{6} (x)

Add a constant to the solution

= \frac{1}{6} cos^{6} (x) + C

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What is the integral of cos^5(x)(-sin(x)) ?
The integral of cos^5(x)(-sin(x)) is 1/6 cos^6(x)+C