What is the integral of sin(4x)sin(10x) ?

The integral of sin(4x)sin(10x) is 1/2 (1/6 sin(6x)-1/14 sin(14x))+C

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integral of sin(4x)sin(10x)

\int sin (4 x) sin (10 x) d x

\frac{1}{2} (\frac{1}{6} sin (6 x) - \frac{1}{14} sin (14 x)) + C

Solution steps

\int sin (4 x) sin (10 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (14 x) + cos (- 6 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 (14 x) + cos (- 6 x) d x

Simplify

- cos (14 x) + cos (- 6 x) : cos (6 x) - cos (14 x)

= \frac{1}{2} \cdot \int cos (6 x) - cos (14 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 (6 x) d x - \int cos (14 x) d x)

\int cos (6 x) d x = \frac{1}{6} sin (6 x)

\int cos (14 x) d x = \frac{1}{14} sin (14 x)

= \frac{1}{2} (\frac{1}{6} sin (6 x) - \frac{1}{14} sin (14 x))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{6} sin (6 x) - \frac{1}{14} sin (14 x)) + C

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\int \frac{1 + sin ^{2} ( x )}{sin ^{2} ( x )} d x

What is the integral of sin(4x)sin(10x) ?
The integral of sin(4x)sin(10x) is 1/2 (1/6 sin(6x)-1/14 sin(14x))+C