integral from 0 to pi of x(pi-x)sin(nx)

解

\int_{0}^{π} x (π - x) sin (n x) d x

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

解答ステップ

\int_{0}^{π} x (π - x) sin (n x) d x

拡張

x (π - x) sin (n x) : π x sin (n x) - x^{2} sin (n x)

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

総和規則を適用する:

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

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

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

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

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