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

解

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

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

解答ステップ

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

部分積分を適用する

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

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

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

限度を計算する:

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

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