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

解答

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

2 x + \frac{6}{5} x^{\frac{5}{6}} + C

求解步骤

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

乘开

\frac{1 + 3 x}{x} : \frac{1}{x} + \frac{1}{6 x}

= \int \frac{1}{x} + \frac{1}{6 x} 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 \frac{1}{x} d x + \int \frac{1}{6 x} d x

\int \frac{1}{x} d x = 2 x

\int \frac{1}{6 x} d x = \frac{6}{5} x^{\frac{5}{6}}

= 2 x + \frac{6}{5} x^{\frac{5}{6}}

解答补常数

= 2 x + \frac{6}{5} x^{\frac{5}{6}} + C

\int csc^{4} (x) cot^{5} (x) d x

\int \frac{8}{36 + x ^{2}} d x

\frac{\partial}{\partial x} (\frac{6 x}{3 x ^{2} + 2 y ^{2} + 9})

\int_{1}^{\infty} 4 e^{- 2 x} d x

\frac{d}{d x} (3 x^{2} + 4 x)