What is the integral of 2sin^7(x)cos(x) ?

The integral of 2sin^7(x)cos(x) is 1/4 sin^8(x)+C

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integral of 2sin^7(x)cos(x)

\int 2 sin^{7} (x) cos (x) d x

\frac{1}{4} sin^{8} (x) + C

Solution steps

\int 2 sin^{7} (x) cos (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int sin^{7} (x) cos (x) d x

Apply u-substitution

: \int u^{7} d u

= 2 \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= 2 \cdot \frac{u ^{8}}{8}

Substitute back

u = sin (x)

= 2 \cdot \frac{sin ^{8} ( x )}{8}

Simplify

2 \cdot \frac{sin ^{8} ( x )}{8} : \frac{1}{4} sin^{8} (x)

= \frac{1}{4} sin^{8} (x)

Add a constant to the solution

= \frac{1}{4} sin^{8} (x) + C

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What is the integral of 2sin^7(x)cos(x) ?
The integral of 2sin^7(x)cos(x) is 1/4 sin^8(x)+C