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)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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}}

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)