What is the integral of sin(pix)sin(2pix) ?

The integral of sin(pix)sin(2pix) is 1/2 (-1/(3pi)sin(3pix)+1/pi sin(pix))+C

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integral of sin(pix)sin(2pix)

\int sin (π x) sin (2 π x) d x

\frac{1}{2} (- \frac{1}{3 π} sin (3 π x) + \frac{1}{π} sin (π x)) + C

Solution steps

\int sin (π x) sin (2 π x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (3 π x) + cos (- π 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 (3 π x) + cos (- π 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 (3 π x) d x + \int cos (- π x) d x)

\int cos (3 π x) d x = \frac{1}{3 π} sin (3 π x)

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

= \frac{1}{2} (- \frac{1}{3 π} sin (3 π x) + \frac{1}{π} sin (π x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{3 π} sin (3 π x) + \frac{1}{π} sin (π x)) + C

What is the integral of sin(pix)sin(2pix) ?
The integral of sin(pix)sin(2pix) is 1/2 (-1/(3pi)sin(3pix)+1/pi sin(pix))+C