What is the integral of sin(4x)sin(5x) ?

The integral of sin(4x)sin(5x) is 1/2 (sin(x)-1/9 sin(9x))+C

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integral of sin(4x)sin(5x)

\int sin (4 x) sin (5 x) d x

\frac{1}{2} (sin (x) - \frac{1}{9} sin (9 x)) + C

Solution steps

\int sin (4 x) sin (5 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (9 x) + cos (- x)) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int - cos (9 x) + cos (- x) d x

Simplify

- cos (9 x) + cos (- x) : cos (x) - cos (9 x)

= \frac{1}{2} \cdot \int cos (x) - cos (9 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{1}{2} (\int cos (x) d x - \int cos (9 x) d x)

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

\int cos (9 x) d x = \frac{1}{9} sin (9 x)

= \frac{1}{2} (sin (x) - \frac{1}{9} sin (9 x))

Add a constant to the solution

= \frac{1}{2} (sin (x) - \frac{1}{9} sin (9 x)) + C

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What is the integral of sin(4x)sin(5x) ?
The integral of sin(4x)sin(5x) is 1/2 (sin(x)-1/9 sin(9x))+C