integral from 0 to 2 of arctan(x/2)

解答

\int_{0}^{2} arctan (\frac{x}{2}) d x

\frac{π}{2} - ln (2)

十进制

0.87764 \dots

求解步骤

\int_{0}^{2} arctan (\frac{x}{2}) d x

使用分布积分法

= [x arctan (\frac{x}{2}) - \int \frac{2 x}{x ^{2} + 4} d x]_{0}^{2}

\int \frac{2 x}{x ^{2} + 4} d x = ln x^{2} + 4

= [x arctan (\frac{x}{2}) - ln x^{2} + 4]_{0}^{2}

带入上下限计算后得:

\frac{π}{2} - ln (2)

= \frac{π}{2} - ln (2)

\int \frac{10}{1 + x ^{2}} d x

x \to 0 lim (\frac{arcsin ( 2 x ) - 2 arcsin ( x )}{x ^{3}})

t an g e n t f (x) = x^{2}

2 x^{2} y \frac{d y}{d x} + 1 + 2 x y^{2} = 0

\frac{\partial}{\partial x} (x y + \frac{27}{x})