What is the integral of 4sin(8x)cos(3x) ?

The integral of 4sin(8x)cos(3x) is 2(-1/11 cos(11x)-1/5 cos(5x))+C

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integral of 4sin(8x)cos(3x)

\int 4 sin (8 x) cos (3 x) d x

2 (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x)) + C

Solution steps

\int 4 sin (8 x) cos (3 x) d x

Take the constant out:

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

= 4 \cdot \int sin (8 x) cos (3 x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= 4 \cdot \int \frac{sin ( 8 x + 3 x ) + sin ( 8 x - 3 x )}{2} d x

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int sin (8 x + 3 x) + sin (8 x - 3 x) d x

Simplify

= 4 \cdot \frac{1}{2} \cdot \int sin (11 x) + sin (5 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot \frac{1}{2} (\int sin (11 x) d x + \int sin (5 x) d x)

\int sin (11 x) d x = - \frac{1}{11} cos (11 x)

\int sin (5 x) d x = - \frac{1}{5} cos (5 x)

= 4 \cdot \frac{1}{2} (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x))

Simplify

4 \cdot \frac{1}{2} (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x)) : 2 (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x))

= 2 (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x))

Add a constant to the solution

= 2 (- \frac{1}{11} cos (11 x) - \frac{1}{5} cos (5 x)) + C

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What is the integral of 4sin(8x)cos(3x) ?
The integral of 4sin(8x)cos(3x) is 2(-1/11 cos(11x)-1/5 cos(5x))+C