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

The integral of-sin(x)cos^4(x) is 1/5 cos^5(x)+C

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

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

\frac{1}{5} cos^{5} (x) + C

Solution steps

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

Simplify

- (- \frac{cos ^{5} ( x )}{5}) : \frac{1}{5} cos^{5} (x)

= \frac{1}{5} cos^{5} (x)

Add a constant to the solution

= \frac{1}{5} cos^{5} (x) + C

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What is the integral of-sin(x)cos^4(x) ?
The integral of-sin(x)cos^4(x) is 1/5 cos^5(x)+C