What is the integral from 0 to 1/2 of xcos(pikx) ?

The integral from 0 to 1/2 of xcos(pikx) is (piksin((pik)/2)+2cos((pik)/2)-2)/(2pi^2k^2)

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integral from 0 to 1/2 of xcos(pikx)

\int_{0}^{\frac{1}{2}} x cos (πk x) d x

\frac{πk sin ( \frac{πk}{2} ) + 2 cos ( \frac{πk}{2} ) - 2}{2 π ^{2} k ^{2}}

Solution steps

\int_{0}^{\frac{1}{2}} x cos (πk x) d x

Apply u-substitution

: \int_{0}^{\frac{πk}{2}} \frac{u cos ( u )}{π ^{2} k ^{2}} d u

= \int_{0}^{\frac{πk}{2}} \frac{u cos ( u )}{π ^{2} k ^{2}} d u

Take the constant out:

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

= \frac{1}{π ^{2} k ^{2}} \cdot \int_{0}^{\frac{πk}{2}} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{\frac{πk}{2}}

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

\int sin (u) d u = - cos (u)

= \frac{1}{π ^{2} k ^{2}} [u sin (u) - (- cos (u))]_{0}^{\frac{πk}{2}}

Simplify

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

Compute the boundaries:

\frac{πk}{2} sin (\frac{πk}{2}) + cos (\frac{πk}{2}) - 1

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

Simplify

= \frac{πk sin ( \frac{πk}{2} ) + 2 cos ( \frac{πk}{2} ) - 2}{2 π ^{2} k ^{2}}

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What is the integral from 0 to 1/2 of xcos(pikx) ?
The integral from 0 to 1/2 of xcos(pikx) is (piksin((pik)/2)+2cos((pik)/2)-2)/(2pi^2k^2)