What is the integral of sin(a+2x)sin(a-2x) ?

The integral of sin(a+2x)sin(a-2x) is 1/2 (-cos(2a)x+1/4 sin(4x))+C

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integral of sin(a+2x)sin(a-2x)

\int sin (a + 2 x) sin (a - 2 x) d x

\frac{1}{2} (- cos (2 a) x + \frac{1}{4} sin (4 x)) + C

Solution steps

\int sin (a + 2 x) sin (a - 2 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (2 a) + cos (4 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 (2 a) + cos (4 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 (2 a) d x + \int cos (4 x) d x)

\int cos (2 a) d x = cos (2 a) x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= \frac{1}{2} (- cos (2 a) x + \frac{1}{4} sin (4 x))

Add a constant to the solution

= \frac{1}{2} (- cos (2 a) x + \frac{1}{4} sin (4 x)) + C

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