integral of xsin((npix)/(10))

解答

\int x sin (\frac{nπ x}{10}) d x

\frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10})) + C

求解步骤

\int x sin (\frac{nπ x}{10}) d x

使用换元积分法

= \int \frac{u sin ( \frac{u}{10} )}{π ^{2} n ^{2}} d u

提出常数:

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

= \frac{1}{π ^{2} n ^{2}} \cdot \int u sin (\frac{u}{10}) d u

使用分布积分法

= \frac{1}{π ^{2} n ^{2}} (- 10 u cos (\frac{u}{10}) - \int - 10 cos (\frac{u}{10}) d u)

\int - 10 cos (\frac{u}{10}) d u = - 100 sin (\frac{u}{10})

= \frac{1}{π ^{2} n ^{2}} (- 10 u cos (\frac{u}{10}) - (- 100 sin (\frac{u}{10})))

u = nπ x

代回

= \frac{1}{π ^{2} n ^{2}} (- 10 nπ x cos (\frac{nπ x}{10}) - (- 100 sin (\frac{nπ x}{10})))

化简

= \frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10}))

解答补常数

= \frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10})) + C

\frac{\partial}{\partial x} (2 x cos (x^{2} + y^{2}))

\frac{d y}{d t} = yr (1000 - y)

a re a y = sec^{2} (x), y = 8 cos (x), - \frac{π}{3}, \frac{π}{3}

\int_{1}^{\infty} \frac{ln ( x )}{x ^{\frac{3}{2}}} d x

\frac{d}{d x} (1000 (1 + \frac{x}{12})^{60})