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

The integral of cos(x)(sin(x))^a is (sin^{a+1}(x))/(a+1)+C

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

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

\frac{sin ^{a + 1} ( x )}{a + 1} + C

Solution steps

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

Simplify

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

Apply u-substitution

: \int u^{a} d u

= \int u^{a} d u

Apply the Power Rule

: \frac{u ^{a + 1}}{a + 1}

= \frac{u ^{a + 1}}{a + 1}

Substitute back

u = sin (x)

= \frac{sin ^{a + 1} ( x )}{a + 1}

Add a constant to the solution

= \frac{sin ^{a + 1} ( x )}{a + 1} + C

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What is the integral of cos(x)(sin(x))^a ?
The integral of cos(x)(sin(x))^a is (sin^{a+1}(x))/(a+1)+C