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

= \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}

Add a constant to the solution

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

\int (x^{2} + 2 x)^{5} (2 x + 2) d x

\frac{d}{d x} (x^{5} e^{- x})

\frac{d}{d x} (\frac{3 x + 5}{x ^{2} - x})

\int \frac{5}{x ln ( 6 x )} d x

\int_{0}^{3} x 16 - x^{2} d x

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