integral of xcos((npix)/l)

解答

\int x cos (\frac{nπ x}{l}) d x

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

求解步骤

\int x cos (\frac{nπ x}{l}) d x

使用分布积分法

= l \frac{1}{πn} x sin (\frac{πn x}{l}) - \int l \frac{1}{πn} sin (\frac{πn x}{l}) d x

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

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

化简

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

解答补常数

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

t a y l or 2^{x}, 0

\int \frac{1}{625 + x ^{2}} d x

\int (x^{2} - 1) e^{2 x} d x

\int \frac{6}{8 - 3 x} d x

t an g e n t f (x) = \frac{4}{x}, at x = - 9