What is the integral of θcos(piθ) ?

The integral of θcos(piθ) is 1/pi θsin(piθ)+1/(pi^2)cos(piθ)+C

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integral of θcos(piθ)

\int θ cos (π θ) d θ

\frac{1}{π} θ sin (π θ) + \frac{1}{π ^{2}} cos (π θ) + C

Solution steps

\int θ cos (π θ) d θ

Apply Integration By Parts

: \frac{1}{π} θ sin (π θ) - \int \frac{1}{π} sin (π θ) d θ

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

\int \frac{1}{π} sin (π θ) d θ = - \frac{1}{π ^{2}} cos (π θ)

= \frac{1}{π} θ sin (π θ) - (- \frac{1}{π ^{2}} cos (π θ))

Simplify

= \frac{1}{π} θ sin (π θ) + \frac{1}{π ^{2}} cos (π θ)

Add a constant to the solution

= \frac{1}{π} θ sin (π θ) + \frac{1}{π ^{2}} cos (π θ) + C

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What is the integral of θcos(piθ) ?
The integral of θcos(piθ) is 1/pi θsin(piθ)+1/(pi^2)cos(piθ)+C