What is the integral of sin(10x)sin(15x) ?

The integral of sin(10x)sin(15x) is 1/2 (1/5 sin(5x)-1/25 sin(25x))+C

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integral of sin(10x)sin(15x)

\int sin (10 x) sin (15 x) d x

\frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{25} sin (25 x)) + C

Solution steps

\int sin (10 x) sin (15 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (25 x) + cos (- 5 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 (25 x) + cos (- 5 x) d x

Simplify

- cos (25 x) + cos (- 5 x) : cos (5 x) - cos (25 x)

= \frac{1}{2} \cdot \int cos (5 x) - cos (25 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 (5 x) d x - \int cos (25 x) d x)

\int cos (5 x) d x = \frac{1}{5} sin (5 x)

\int cos (25 x) d x = \frac{1}{25} sin (25 x)

= \frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{25} sin (25 x))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{25} sin (25 x)) + C

\frac{d}{d x} (\frac{ln ( x )}{x ^{11}})

\frac{x ^{3}}{x ^{2} - 1}

k = 1 \sum \infty 0. 5^{k}

\int \frac{1}{4096} cos (8 x) d x

What is the integral of sin(10x)sin(15x) ?
The integral of sin(10x)sin(15x) is 1/2 (1/5 sin(5x)-1/25 sin(25x))+C