What is the integral of pi/(20)sin(5x)cos(8x) ?

The integral of pi/(20)sin(5x)cos(8x) is pi/(40)(-1/13 cos(13x)+1/3 cos(3x))+C

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integral of pi/(20)sin(5x)cos(8x)

\int \frac{π}{20} sin (5 x) cos (8 x) d x

\frac{π}{40} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x)) + C

Solution steps

\int \frac{π}{20} sin (5 x) cos (8 x) d x

Take the constant out:

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

= \frac{π}{20} \cdot \int sin (5 x) cos (8 x) d x

Use the following identity:

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

= \frac{π}{20} \cdot \int \frac{sin ( 5 x + 8 x ) + sin ( 5 x - 8 x )}{2} d x

Take the constant out:

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

= \frac{π}{20} \cdot \frac{1}{2} \cdot \int sin (5 x + 8 x) + sin (5 x - 8 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

= \frac{π}{20} \cdot \frac{1}{2} (\int sin (5 x + 8 x) d x + \int sin (5 x - 8 x) d x)

\int sin (5 x + 8 x) d x = - \frac{1}{13} cos (13 x)

\int sin (5 x - 8 x) d x = \frac{1}{3} cos (3 x)

= \frac{π}{20} \cdot \frac{1}{2} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x))

Simplify

\frac{π}{20} \cdot \frac{1}{2} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x)) : \frac{π}{40} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x))

= \frac{π}{40} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x))

Add a constant to the solution

= \frac{π}{40} (- \frac{1}{13} cos (13 x) + \frac{1}{3} cos (3 x)) + C

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What is the integral of pi/(20)sin(5x)cos(8x) ?
The integral of pi/(20)sin(5x)cos(8x) is pi/(40)(-1/13 cos(13x)+1/3 cos(3x))+C