What is the integral of sin(piw)cos(piw) ?

The integral of sin(piw)cos(piw) is -1/(4pi)cos(2piw)+C

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integral of sin(piw)cos(piw)

\int sin (π w) cos (π w) d w

- \frac{1}{4 π} cos (2 π w) + C

Solution steps

\int sin (π w) cos (π w) d w

Rewrite using trig identities

= \int \frac{sin ( 2 π w )}{2} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin (2 π w) d w

Apply u-substitution

: \int sin (u) \frac{1}{2 π} d u

= \frac{1}{2} \cdot \int sin (u) \frac{1}{2 π} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \frac{1}{2 π} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= \frac{1}{2} \cdot \frac{1}{2 π} (- cos (u))

Substitute back

u = 2 π w

= \frac{1}{2} \cdot \frac{1}{2 π} (- cos (2 π w))

Simplify

\frac{1}{2} \cdot \frac{1}{2 π} (- cos (2 π w)) : - \frac{1}{4 π} cos (2 π w)

= - \frac{1}{4 π} cos (2 π w)

Add a constant to the solution

= - \frac{1}{4 π} cos (2 π w) + C

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What is the integral of sin(piw)cos(piw) ?
The integral of sin(piw)cos(piw) is -1/(4pi)cos(2piw)+C