What is the integral of (sin(x))^ncos(x) ?

The integral of (sin(x))^ncos(x) is (sin^{n+1}(x))/(n+1)+C

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integral of (sin(x))^ncos(x)

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

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

Solution steps

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

Simplify

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

Apply u-substitution

: \int u^{n} d u

= \int u^{n} d u

Apply the Power Rule

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

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

Substitute back

u = sin (x)

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

Add a constant to the solution

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

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