What is the integral of 8cos(8x)sin(8x) ?

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

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integral of 8cos(8x)sin(8x)

\int 8 cos (8 x) sin (8 x) d x

- \frac{1}{2} cos^{2} (8 x) + C

Solution steps

\int 8 cos (8 x) sin (8 x) d x

Take the constant out:

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

= 8 \cdot \int cos (8 x) sin (8 x) d x

Apply u-substitution

: \int - \frac{u}{8} d u

= 8 \cdot \int - \frac{u}{8} d u

Take the constant out:

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

= 8 (- \frac{1}{8} \cdot \int u d u)

Apply the Power Rule

: \frac{u ^{2}}{2}

= 8 (- \frac{1}{8} \cdot \frac{u ^{2}}{2})

Substitute back

u = cos (8 x)

= 8 (- \frac{1}{8} \cdot \frac{cos ^{2} ( 8 x )}{2})

Simplify

8 (- \frac{1}{8} \cdot \frac{cos ^{2} ( 8 x )}{2}) : - \frac{1}{2} cos^{2} (8 x)

= - \frac{1}{2} cos^{2} (8 x)

Add a constant to the solution

= - \frac{1}{2} cos^{2} (8 x) + C

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