integral of x(l-x)sin((npix)/l)

解

\int x (l - x) sin (\frac{nπ x}{l}) d x

- l^{2} \frac{1}{πn} x cos (\frac{πn x}{l}) + l^{2} \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}) + l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) - 2 l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) - 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l}) + C

解答ステップ

\int x (l - x) sin (\frac{nπ x}{l}) d x

部分積分を適用する

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

\int l \frac{1}{πn} x cos (\frac{πn x}{l}) - l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}) d x = l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) + 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l})

= (l - x) (- l \frac{1}{πn} x cos (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l})) - (l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) + 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l}))

簡素化

(l - x) (- l \frac{1}{πn} x cos (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l})) - (l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) + 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l})) : - l^{2} \frac{1}{πn} x cos (\frac{πn x}{l}) + l^{2} \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}) + l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) - 2 l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) - 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l})

= - l^{2} \frac{1}{πn} x cos (\frac{πn x}{l}) + l^{2} \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}) + l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) - 2 l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) - 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l})

定数を解答に追加する

= - l^{2} \frac{1}{πn} x cos (\frac{πn x}{l}) + l^{2} \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}) + l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) - 2 l \frac{l}{π ^{2} n ^{2}} x sin (\frac{πn x}{l}) - 2 l \frac{l ^{2}}{π ^{3} n ^{3}} cos (\frac{πn x}{l}) + C