What is the integral from 0 to x^2 of xcos(pix) ?

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

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integral from 0 to x^2 of xcos(pix)

\int_{0}^{x^{2}} x cos (π x) d x

\frac{x ^{2} sin ( π x ^{2} )}{π} + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}

Solution steps

\int_{0}^{x^{2}} 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}^{x^{2}}

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

\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}^{x^{2}}

Simplify

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

Compute the boundaries:

\frac{1}{π} x^{2} sin (π x^{2}) + \frac{1}{π ^{2}} cos (π x^{2}) - \frac{1}{π ^{2}}

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

Simplify

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

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