What is the integral of sin(4x-2)sin(3x+5) ?

The integral of sin(4x-2)sin(3x+5) is 1/2 (-1/7 sin(7x+3)+sin(x-7))+C

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integral of sin(4x-2)sin(3x+5)

\int sin (4 x - 2) sin (3 x + 5) d x

\frac{1}{2} (- \frac{1}{7} sin (7 x + 3) + sin (x - 7)) + C

Solution steps

\int sin (4 x - 2) sin (3 x + 5) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (7 x + 3) + cos (x - 7)) 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 (7 x + 3) + cos (x - 7) 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 (7 x + 3) d x + \int cos (x - 7) d x)

\int cos (7 x + 3) d x = \frac{1}{7} sin (7 x + 3)

\int cos (x - 7) d x = sin (x - 7)

= \frac{1}{2} (- \frac{1}{7} sin (7 x + 3) + sin (x - 7))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{7} sin (7 x + 3) + sin (x - 7)) + C

4 cos (3 x), 4 - 4 cos (3 x), [0, \frac{π}{3}]

- 2 x^{3} - 4 x^{2}, at x = 1

\int cot^{2} (x) csc^{4} (x) d x

\frac{\partial}{\partial x} (sin (3 x + 2 y))

\frac{s}{( s ^{2} + 16 ) ^{2}}

What is the integral of sin(4x-2)sin(3x+5) ?
The integral of sin(4x-2)sin(3x+5) is 1/2 (-1/7 sin(7x+3)+sin(x-7))+C