What is the integral from 0 to 1 of (x-x^2)sin(npix) ?

The integral from 0 to 1 of (x-x^2)sin(npix) is -(2(-1)^n-2)/(pi^3n^3)

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integral from 0 to 1 of (x-x^2)sin(npix)

\int_{0}^{1} (x - x^{2}) sin (nπ x) d x

- \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}}

Solution steps

\int_{0}^{1} (x - x^{2}) sin (nπ x) d x

Expand

(x - x^{2}) sin (nπ x) : x sin (πn x) - x^{2} sin (πn x)

= \int_{0}^{1} x sin (πn x) - x^{2} sin (πn x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} x sin (πn x) d x - \int_{0}^{1} x^{2} sin (πn x) d x

\int_{0}^{1} x sin (πn x) d x = - \frac{( - 1 ) ^{n}}{πn}

\int_{0}^{1} x^{2} sin (πn x) d x = \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn}

= - \frac{( - 1 ) ^{n}}{πn} - (\frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn})

Simplify

= - \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}}

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\int_{0}^{1} (\frac{x - 1}{x + 1})^{3} d x

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What is the integral from 0 to 1 of (x-x^2)sin(npix) ?
The integral from 0 to 1 of (x-x^2)sin(npix) is -(2(-1)^n-2)/(pi^3n^3)