integral of (3/(x^{3/4)}-4\sqrt[3]{x}+1)

解答

\int (\frac{3}{x ^{\frac{3}{4}}} - 4 3 x + 1) d x

12 4 x - 3 x^{\frac{4}{3}} + x + C

求解步骤

\int \frac{3}{x ^{\frac{3}{4}}} - 4 3 x + 1 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{3}{x ^{\frac{3}{4}}} d x - \int 4 3 x d x + \int 1 d x

\int \frac{3}{x ^{\frac{3}{4}}} d x = 12 x^{\frac{1}{4}}

\int 4 3 x d x = 3 x^{\frac{4}{3}}

\int 1 d x = x

= 12 x^{\frac{1}{4}} - 3 x^{\frac{4}{3}} + x

使用根式运算法则:

a^{\frac{1}{n}} = n a

= 12 4 x - 3 x^{\frac{4}{3}} + x

解答补常数

= 12 4 x - 3 x^{\frac{4}{3}} + x + C

\frac{d y}{d x} = (x - 2) e^{- 2 y}

\frac{\partial}{\partial x} ((2 - x) (5 - x - 2 y))

x \to 2 lim (\frac{3 x + 4}{5 x + 7})

\frac{d}{d x} (\frac{1}{cos ( x )} + \frac{1}{sin ( x + 1 )})

\frac{d}{d θ} (\frac{e ^{i θ} + e ^{- i θ}}{2})