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

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

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

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

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

Solution steps

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

f (x) = 5 ln (4 x)

f (x) = \frac{x - 5}{( x + 3 ) ^{4}}

\int sin (x) sec^{10} (x) d x

\frac{8}{t ^{8}} - \frac{6}{t ^{6}} + \frac{3}{t}

f (x) = arctan (sin (4 x))

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