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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

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

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