What is the integral of sin(1-2x)sin(2x+3) ?

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

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integral of sin(1-2x)sin(2x+3)

\int sin (1 - 2 x) sin (2 x + 3) d x

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

Solution steps

\int sin (1 - 2 x) sin (2 x + 3) d x

Rewrite using trig identities

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

\int cos (4) d x = cos (4) x

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

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

Add a constant to the solution

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

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