integral from 0 to L of x^2sin((npix)/L)

解

\int_{0}^{L} x^{2} sin (\frac{nπ x}{L}) d x

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

解答ステップ

\int_{0}^{L} x^{2} sin (\frac{nπ x}{L}) d x

部分積分を適用する

= [- L \frac{1}{πn} x^{2} cos (\frac{πn x}{L}) - \int - L \frac{2}{πn} x cos (\frac{πn x}{L}) d x]_{0}^{L}

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

= [- L \frac{1}{πn} x^{2} cos (\frac{πn x}{L}) - (- \frac{2 L}{πn} (L \frac{1}{πn} x sin (\frac{πn x}{L}) + L \frac{L}{π ^{2} n ^{2}} cos (\frac{πn x}{L})))]_{0}^{L}

簡素化

= [- L \frac{1}{πn} x^{2} cos (\frac{πn x}{L}) + \frac{2 L}{πn} (L \frac{1}{πn} x sin (\frac{πn x}{L}) + L \frac{L}{π ^{2} n ^{2}} cos (\frac{πn x}{L}))]_{0}^{L}

限度を計算する:

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

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

簡素化

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