What is the integral from 0 to 1 of xcos(pix) ?

The integral from 0 to 1 of xcos(pix) is -2/(pi^2)

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integral from 0 to 1 of xcos(pix)

\int_{0}^{1} x cos (π x) d x

- \frac{2}{π ^{2}}

Decimal

- 0.20264 \dots

Solution steps

\int_{0}^{1} x cos (π x) d x

Apply Integration By Parts

= [\frac{1}{π} x sin (π x) - \int \frac{1}{π} sin (π x) d x]_{0}^{1}

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

= [\frac{1}{π} x sin (π x) - (- \frac{1}{π ^{2}} cos (π x))]_{0}^{1}

Simplify

= [\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x)]_{0}^{1}

Compute the boundaries:

- \frac{2}{π ^{2}}

= - \frac{2}{π ^{2}}

What is the integral from 0 to 1 of xcos(pix) ?
The integral from 0 to 1 of xcos(pix) is -2/(pi^2)