What is the integral of x*sin((npix)/2) ?

The integral of x*sin((npix)/2) is 4/(pi^2n^2)(-(pin)/2 xcos((pinx)/2)+sin((pinx)/2))+C

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integral of x*sin((npix)/2)

\int x \cdot sin (\frac{nπ x}{2}) d x

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

Solution steps

\int x sin (\frac{nπ x}{2}) d x

Apply u-substitution

: \int \frac{4 u sin ( u )}{π ^{2} n ^{2}} d u

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

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

= \frac{4}{π ^{2} n ^{2}} (- u cos (u) - \int - cos (u) d u)

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

= \frac{4}{π ^{2} n ^{2}} (- u cos (u) - (- sin (u)))

Substitute back

u = \frac{n x π}{2}

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (x^{3} + y^{3})

\int 3 x sin (x^{2}) d x

y^{^{'''}} + 6 y^{^{''}} + 9 y^{^{'}} = 0

\int_{1}^{6} \frac{2}{x - 11} d x

\frac{d x}{d t} + 4 \cdot 1 0^{- 4} x = 8 \cdot 1 0^{- 5}, x (0) = 1

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