integral of sqrt(x)(sqrt(a)-sqrt(x))^2

解答

\int x (a - x)^{2} d x

a \frac{2}{3} x^{\frac{3}{2}} - a x^{2} + \frac{2}{5} x^{\frac{5}{2}} + C

求解步骤

\int x (a - x)^{2} d x

乘开

x (a - x)^{2} : a x - 2 a x + x^{\frac{3}{2}}

= \int a x - 2 a x + x^{\frac{3}{2}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int a x d x - \int 2 a x d x + \int x^{\frac{3}{2}} d x

\int a x d x = a \frac{2}{3} x^{\frac{3}{2}}

\int 2 a x d x = a x^{2}

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

= a \frac{2}{3} x^{\frac{3}{2}} - a x^{2} + \frac{2}{5} x^{\frac{5}{2}}

解答补常数

= a \frac{2}{3} x^{\frac{3}{2}} - a x^{2} + \frac{2}{5} x^{\frac{5}{2}} + C

\frac{d}{d x} (\frac{2}{x ^{2} + 1})

\int_{0}^{2} 2 π (5 - x) (8 - x^{3}) d x

\int (\frac{4}{x ^{2}}) d x

\frac{x d y}{d x} + 5 y = 7 x^{2}, f (2) = 5

\frac{d}{d x} (C x e^{- x})