What is the integral of sin(5x)sin(3x) ?

The integral of sin(5x)sin(3x) is 1/2 (-1/8 sin(8x)+1/2 sin(2x))+C

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integral of sin(5x)sin(3x)

\int sin (5 x) sin (3 x) d x

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

Solution steps

\int sin (5 x) sin (3 x) d x

Rewrite using trig identities

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

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

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

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

Add a constant to the solution

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

\int \frac{48 x ^{2}}{x ^{4} - 74 x ^{2} + 1225} d x

- \frac{1}{s}

x \to 0 lim (1 - x^{2})

\frac{d y}{d x} = \frac{( y - 1 )}{( x + 3 )}

x \to - 2 lim (\frac{x ^{3} - 8}{x + 2})

What is the integral of sin(5x)sin(3x) ?
The integral of sin(5x)sin(3x) is 1/2 (-1/8 sin(8x)+1/2 sin(2x))+C