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

The integral from 0 to 2 of xcos(pix) is 0

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integral from 0 to 2 of xcos(pix)

\int_{0}^{2} x cos (π x) d x

0

Solution steps

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

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

Simplify

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

Compute the boundaries:

0

= 0

\int_{0}^{26} x^{2} - 13 d x

\int_{0}^{2} (2 x - 3) (4 x^{2} + 4) d x

\int_{0}^{2} \frac{x ^{3}}{16 - x ^{2}} d x

\int_{- 4}^{4} 2 x d x

\int_{1}^{2} 2 x (x^{2} + 6) d x

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