integral of x/(\sqrt[3]{x-1)}

解答

\int \frac{x}{3 x - 1} d x

\frac{3}{5} (x - 1)^{\frac{5}{3}} + \frac{3}{2} (x - 1)^{\frac{2}{3}} + C

求解步骤

\int \frac{x}{3 x - 1} d x

使用换元积分法

= \int \frac{u + 1}{3 u} d u

乘开

\frac{u + 1}{3 u} : u^{\frac{2}{3}} + \frac{1}{3 u}

= \int u^{\frac{2}{3}} + \frac{1}{3 u} d u

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 u^{\frac{2}{3}} d u + \int \frac{1}{3 u} d u

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

\int \frac{1}{3 u} d u = \frac{3}{2} u^{\frac{2}{3}}

= \frac{3}{5} u^{\frac{5}{3}} + \frac{3}{2} u^{\frac{2}{3}}

u = x - 1

代回

= \frac{3}{5} (x - 1)^{\frac{5}{3}} + \frac{3}{2} (x - 1)^{\frac{2}{3}}

解答补常数

= \frac{3}{5} (x - 1)^{\frac{5}{3}} + \frac{3}{2} (x - 1)^{\frac{2}{3}} + C

\int x e^{- \frac{1}{10} x^{2}} d x

\frac{d ^{3}}{d x ^{3}} ((1 + x)^{3})

\frac{\partial}{\partial x} (2 x y^{3} + 2)

y^{^{'}} - (\frac{3}{t}) \cdot y = 2 t^{3} \cdot e^{(2 t)}, y (1) = 0

n = 1 \sum \infty \frac{1}{2 ^{n} - 5}