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

The integral of cos^7(x)sin(2x) is -2/9 cos^9(x)+C

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

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

- \frac{2}{9} cos^{9} (x) + C

Solution steps

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

Rewrite using trig identities

= \int 2 cos^{8} (x) sin (x) d x

Take the constant out:

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

= 2 \cdot \int cos^{8} (x) sin (x) d x

Apply u-substitution

: \int - u^{8} d u

= 2 \cdot \int - u^{8} d u

Take the constant out:

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

= 2 (- \int u^{8} d u)

Apply the Power Rule

: \frac{u ^{9}}{9}

= 2 (- \frac{u ^{9}}{9})

Substitute back

u = cos (x)

= 2 (- \frac{cos ^{9} ( x )}{9})

Simplify

2 (- \frac{cos ^{9} ( x )}{9}) : - \frac{2}{9} cos^{9} (x)

= - \frac{2}{9} cos^{9} (x)

Add a constant to the solution

= - \frac{2}{9} cos^{9} (x) + C

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What is the integral of cos^7(x)sin(2x) ?
The integral of cos^7(x)sin(2x) is -2/9 cos^9(x)+C