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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of sin(pix)cos(2pix)

\int sin (π x) cos (2 π x) d x

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

Solution steps

\int sin (π x) cos (2 π x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int \frac{sin ( π x + 2 π x ) + sin ( π x - 2 π x )}{2} 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 sin (π x + 2 π x) + sin (π x - 2 π x) d x

Simplify

sin (π x + 2 π x) + sin (π x - 2 π x) : sin (3 π x) - sin (π x)

= \frac{1}{2} \cdot \int sin (3 π x) - sin (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 sin (3 π x) d x - \int sin (π x) d x)

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

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

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

Simplify

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

Add a constant to the solution

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

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