What is the integral of 2xsin(npix) ?

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

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integral of 2xsin(npix)

\int 2 x sin (nπ x) d x

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

Solution steps

\int 2 x sin (nπ x) d x

Take the constant out:

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

= 2 \cdot \int x sin (nπ x) d x

Apply Integration By Parts

= 2 (- \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)

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

Simplify

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

Add a constant to the solution

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

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