What is the integral of xsin(npix) ?

The integral of xsin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)+C

integral of xsin(npix)

\int x sin (nπ x) d x

- \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x) + C

Solution steps

\int x sin (nπ x) d x

Apply Integration By Parts

= - \frac{1}{πn} x cos (πn x) - \int - \frac{1}{πn} cos (πn x) d x

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

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

Simplify

= - \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x)

Add a constant to the solution

= - \frac{1}{πn} x cos (πn x) + \frac{1}{π ^{2} n ^{2}} sin (πn x) + C

\int_{0}^{x} t (e^{t^{3}} - 1) d t

\int cos^{3} (3 θ) sin^{- 2} (3 θ) d θ

d er i v a t i v e \frac{2 t + 1}{t + 3}

d er i v a t i v e 8 x + 9 x e^{x}

\int cos (\frac{nπ x}{L}) d x

What is the integral of xsin(npix) ?
The integral of xsin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)+C