What is the integral of xcos(n*pi*x/2) ?

The integral of xcos(n*pi*x/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 (n \cdot π \cdot \frac{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 (nπ \frac{x}{2}) d x

Multiply

nπ \frac{x}{2} : \frac{πn x}{2}

= \int x cos (\frac{πn x}{2}) d x

Apply u-substitution

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

= \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

: \int 4 v cos (v) d v

= \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

: v sin (v) - \int sin (v) d v

= \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{πn x}{2} sin (\frac{πn x}{2}) - (- cos (\frac{πn x}{2})))

Simplify

\frac{1}{π ^{2} n ^{2}} \cdot 4 (\frac{πn x}{2} 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}))

= \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

\frac{d}{d x} (\frac{1}{2 x - 1})

\int \frac{12 x}{2 x - 5} d x

x \to 0 lim (\frac{csc ( 2 x )}{x})

\frac{d}{d x} ((3 x - 5)^{7} (5 x + 2)^{4})

\frac{\partial}{\partial x} (t^{2} e^{\frac{x ^{2}}{4 t}})

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