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

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

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

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

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

Solution steps

\int sin (5 x) sin (4 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

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 (9 x) d x + \int cos (x) d x)

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

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

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

Add a constant to the solution

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

(4.1) (8.2)

\frac{d}{d x} (\frac{sec ( x ) + csc ( x )}{csc ( x )})

\int cos (x) (tan (x) + sec (x)) d x

\frac{\partial}{\partial x} (- x^{2} y + 4)

\frac{d}{d x} (25 - x^{2} - y^{2} - z^{2})

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