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

解答

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

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

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

使用分布积分法

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

\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})))]_{- L}^{0}

化简

= - [- 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}))]_{- L}^{0}

带入上下限计算后得:

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

化简

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

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