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

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

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integral of x*sin(npix)

\int x \cdot 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

= - \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

\frac{d}{d θ} (2 + 2 sin (θ))

\int \frac{x}{49 + x ^{2}} d x

\int cos^{2} (π t) sin (π t) d t

\frac{d}{d x} (\frac{3 x}{x + 1})

x^{2} e^{- 7 x}

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