What is the integral of sin(2x)sin(7x) ?

The integral of sin(2x)sin(7x) is 1/2 (1/5 sin(5x)-1/9 sin(9x))+C

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integral of sin(2x)sin(7x)

\int sin (2 x) sin (7 x) d x

\frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x)) + C

Solution steps

\int sin (2 x) sin (7 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (9 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 (9 x) + cos (- 5 x) d x

Simplify

- cos (9 x) + cos (- 5 x) : cos (5 x) - cos (9 x)

= \frac{1}{2} \cdot \int cos (5 x) - cos (9 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 (9 x) d x)

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

\int cos (9 x) d x = \frac{1}{9} sin (9 x)

= \frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{5} sin (5 x) - \frac{1}{9} sin (9 x)) + C

\frac{d y}{d x} = \frac{sin ( x )}{y}, y (0) = 1

\frac{\partial}{\partial x} (y \cdot e^{- x})

(3 x^{3})^{^{'}}

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

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

What is the integral of sin(2x)sin(7x) ?
The integral of sin(2x)sin(7x) is 1/2 (1/5 sin(5x)-1/9 sin(9x))+C