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

解答

\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

求解步骤

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

使用分布积分法

= (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}))

化简

(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})

解答补常数

= - 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

\int (x^{2} - 2 x + 4) e^{- 4 x + 5} d x

\int \frac{x ^{2}}{2 x + 12} d x

\int \frac{x + 1}{( x - 1 ) ( x + 2 )} d x

\int sin (x) \cdot e^{2 x} d x

\int (\frac{e ^{2 x} - e ^{x}}{e ^{x}}) d x