What is the integral of sin(6x)sin(2x) ?

The integral of sin(6x)sin(2x) is 1/2 (-1/8 sin(8x)+1/4 sin(4x))+C

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integral of sin(6x)sin(2x)

\int sin (6 x) sin (2 x) d x

\frac{1}{2} (- \frac{1}{8} sin (8 x) + \frac{1}{4} sin (4 x)) + C

Solution steps

\int sin (6 x) sin (2 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (8 x) + 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 (8 x) + 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 (8 x) d x + \int cos (4 x) d x)

\int cos (8 x) d x = \frac{1}{8} sin (8 x)

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

= \frac{1}{2} (- \frac{1}{8} sin (8 x) + \frac{1}{4} sin (4 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{8} sin (8 x) + \frac{1}{4} sin (4 x)) + C

y^{^{''}} + 6 y^{^{'}} + 8 y = 0

\int x^{- 4} + x^{\frac{1}{3}} d x

s (t) = 3 t + 1

\frac{\partial}{\partial y} (x^{2} yz - x y z^{3})

\frac{d}{d y} (\frac{y}{x})

What is the integral of sin(6x)sin(2x) ?
The integral of sin(6x)sin(2x) is 1/2 (-1/8 sin(8x)+1/4 sin(4x))+C