What is the integral from 0 to 2 of (5xcos(pix)) ?

The integral from 0 to 2 of (5xcos(pix)) is 0

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integral from 0 to 2 of (5xcos(pix))

\int_{0}^{2} (5 x cos (π x)) d x

0

Solution steps

\int_{0}^{2} 5 x cos (π x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \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}

= 5 [\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)

= 5 [\frac{1}{π} x sin (π x) - (- \frac{1}{π ^{2}} cos (π x))]_{0}^{2}

Simplify

= 5 [\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x)]_{0}^{2}

Compute the boundaries:

0

= 5 \cdot 0

Simplify

= 0

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What is the integral from 0 to 2 of (5xcos(pix)) ?
The integral from 0 to 2 of (5xcos(pix)) is 0