What is the integral of-4sin(3x)sin(nx) ?

The integral of-4sin(3x)sin(nx) is -2(-1/(3+n)sin(3x+nx)+1/(3-n)sin(3x-nx))+C

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integral of-4sin(3x)sin(nx)

\int - 4 sin (3 x) sin (n x) d x

- 2 (- \frac{1}{3 + n} sin (3 x + n x) + \frac{1}{3 - n} sin (3 x - n x)) + C

Solution steps

\int - 4 sin (3 x) sin (n 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 (3 x) sin (n x) d x

Rewrite using trig identities

= - 4 \cdot \int \frac{- cos ( 3 x + n x ) + cos ( 3 x - n 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 - cos (3 x + n x) + cos (3 x - n 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 cos (3 x + n x) d x + \int cos (3 x - n x) d x)

\int cos (3 x + n x) d x = \frac{1}{3 + n} sin (3 x + n x)

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

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

Simplify

- 4 \cdot \frac{1}{2} (- \frac{1}{3 + n} sin (3 x + n x) + \frac{1}{3 - n} sin (3 x - n x)) : - 2 (- \frac{1}{3 + n} sin (3 x + n x) + \frac{1}{3 - n} sin (3 x - n x))

= - 2 (- \frac{1}{3 + n} sin (3 x + n x) + \frac{1}{3 - n} sin (3 x - n x))

Add a constant to the solution

= - 2 (- \frac{1}{3 + n} sin (3 x + n x) + \frac{1}{3 - n} sin (3 x - n x)) + C

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What is the integral of-4sin(3x)sin(nx) ?
The integral of-4sin(3x)sin(nx) is -2(-1/(3+n)sin(3x+nx)+1/(3-n)sin(3x-nx))+C