What is the integral from 0 to 0.5 of xcos(pix) ?

The integral from 0 to 0.5 of xcos(pix) is 0.05783…

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integral from 0 to 0.5 of xcos(pix)

\int_{0}^{0.5} x cos (π x) d x

0.05783 \dots

Solution steps

\int_{0}^{0.5} x cos (π x) d x

Apply Integration By Parts

: [\frac{1}{π} (x sin (π x) - π \cdot \int \frac{1}{π} sin (π x) d x)]_{0}^{0.5}

= [\frac{1}{π} (x sin (π x) - π \cdot \int \frac{1}{π} sin (π x) d x)]_{0}^{0.5}

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

= [\frac{1}{π} (x sin (π x) - π (- \frac{1}{π ^{2}} cos (π x)))]_{0}^{0.5}

Simplify

[\frac{1}{π} (x sin (π x) - π (- \frac{1}{π ^{2}} cos (π x)))]_{0}^{0.5} : [\frac{1}{π} (x sin (π x) + \frac{1}{π} cos (π x))]_{0}^{0.5}

= [\frac{1}{π} (x sin (π x) + \frac{1}{π} cos (π x))]_{0}^{0.5}

Compute the boundaries:

0.05783 \dots

= 0.05783 \dots

\int_{2}^{3} x + 3 d x

\int_{4}^{6} (\frac{5}{3} x - 5) d x

\int_{- 1}^{5} 4 x^{3} - 3 x^{2} + 1 d x

\int_{1}^{4} \frac{1}{( x - 3 ) ^{\frac{1}{5}}} d x

\int_{0}^{\frac{π}{6}} x cos (x + \frac{π}{3}) d x

What is the integral from 0 to 0.5 of xcos(pix) ?
The integral from 0 to 0.5 of xcos(pix) is 0.05783…