What is the integral of xcos((npix)/2) ?

The integral of xcos((npix)/2) is 4/(pi^2n^2)((pin)/2 xsin((pinx)/2)+cos((pinx)/2))+C

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integral of xcos((npix)/2)

\int x cos (\frac{nπ x}{2}) d x

\frac{4}{π ^{2} n ^{2}} (\frac{πn}{2} x sin (\frac{πn x}{2}) + cos (\frac{πn x}{2})) + C

Solution steps

\int x cos (\frac{nπ x}{2}) d x

Apply u-substitution

= \int \frac{u cos ( \frac{u}{2} )}{π ^{2} n ^{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} n ^{2}} \cdot \int u cos (\frac{u}{2}) d u

Apply u-substitution

= \frac{1}{π ^{2} n ^{2}} \cdot \int 4 v cos (v) d v

Take the constant out:

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

= \frac{1}{π ^{2} n ^{2}} \cdot 4 \cdot \int v cos (v) d v

Apply Integration By Parts

= \frac{1}{π ^{2} n ^{2}} \cdot 4 (v sin (v) - \int sin (v) d v)

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

= \frac{1}{π ^{2} n ^{2}} \cdot 4 (v sin (v) - (- cos (v)))

Substitute back

= \frac{1}{π ^{2} n ^{2}} \cdot 4 (\frac{nπ x}{2} sin (\frac{nπ x}{2}) - (- cos (\frac{nπ x}{2})))

Simplify

\frac{1}{π ^{2} n ^{2}} \cdot 4 (\frac{nπ x}{2} sin (\frac{nπ x}{2}) - (- cos (\frac{nπ x}{2}))) : \frac{4}{π ^{2} n ^{2}} (\frac{πn}{2} x sin (\frac{πn x}{2}) + cos (\frac{πn x}{2}))

= \frac{4}{π ^{2} n ^{2}} (\frac{πn}{2} x sin (\frac{πn x}{2}) + cos (\frac{πn x}{2}))

Add a constant to the solution

= \frac{4}{π ^{2} n ^{2}} (\frac{πn}{2} x sin (\frac{πn x}{2}) + cos (\frac{πn x}{2})) + C

What is the integral of xcos((npix)/2) ?
The integral of xcos((npix)/2) is 4/(pi^2n^2)((pin)/2 xsin((pinx)/2)+cos((pinx)/2))+C