What is the integral of xcos((pix)/4) ?

The integral of xcos((pix)/4) is 4/pi xsin((pix)/4)+(16}{pi^2}cos(\frac{pix)/4)+C

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integral of xcos((pix)/4)

\int x cos (\frac{π x}{4}) d x

\frac{4}{π} x sin (\frac{π x}{4}) + \frac{16}{π ^{2}} cos (\frac{π x}{4}) + C

Solution steps

\int x cos (\frac{π x}{4}) d x

Apply Integration By Parts

: \frac{4}{π} x sin (\frac{π x}{4}) - \int \frac{4}{π} sin (\frac{π x}{4}) d x

= \frac{4}{π} x sin (\frac{π x}{4}) - \int \frac{4}{π} sin (\frac{π x}{4}) d x

\int \frac{4}{π} sin (\frac{π x}{4}) d x = - \frac{16}{π ^{2}} cos (\frac{π x}{4})

= \frac{4}{π} x sin (\frac{π x}{4}) - (- \frac{16}{π ^{2}} cos (\frac{π x}{4}))

Simplify

= \frac{4}{π} x sin (\frac{π x}{4}) + \frac{16}{π ^{2}} cos (\frac{π x}{4})

Add a constant to the solution

= \frac{4}{π} x sin (\frac{π x}{4}) + \frac{16}{π ^{2}} cos (\frac{π x}{4}) + C

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What is the integral of xcos((pix)/4) ?
The integral of xcos((pix)/4) is 4/pi xsin((pix)/4)+(16}{pi^2}cos(\frac{pix)/4)+C