What is the integral from 1/2 to 1 of xcos(pix) ?

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

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integral from 1/2 to 1 of xcos(pix)

\int_{\frac{1}{2}}^{1} x cos (π x) d x

- \frac{1}{π ^{2}} - \frac{1}{2 π}

Decimal Notation

- 0.26047 \dots

Solution steps

\int_{\frac{1}{2}}^{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]_{\frac{1}{2}}^{1}

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

Simplify

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

Compute the boundaries:

- \frac{1}{π ^{2}} - \frac{1}{2 π}

= - \frac{1}{π ^{2}} - \frac{1}{2 π}

\int_{0}^{5} \frac{1}{9 + 4 x ^{2}} d x

x \to 6 lim (3 x)

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

\int (\frac{1}{x ^{8}}) d x

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