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受欢迎的微积分 >

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

解答

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

解答

- \frac{3}{2} ln 1 - x^{\frac{2}{3}} + C

求解步骤

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

使用换元积分法

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

使用换元积分法

= 3 \cdot \int - \frac{1}{2 v} d v

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 (- \frac{1}{2} \cdot \int \frac{1}{v} d v)

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 (- \frac{1}{2} ln ∣ v ∣)

代回

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

化简

3 (- \frac{1}{2} ln 1 - (3 x)^{2}) : - \frac{3}{2} ln 1 - x^{\frac{2}{3}}

= - \frac{3}{2} ln 1 - x^{\frac{2}{3}}

解答补常数

= - \frac{3}{2} ln 1 - x^{\frac{2}{3}} + C

作图

流行的例子

tangent (8x^2+4x+4)/(sqrt(4))

t an g e n t \frac{8 x ^{2} + 4 x + 4}{4}

integral of (e^x+5)

\int (e^{x} + 5) d x

limit as x approaches 3 of 9-x^2

x \to 3 lim (9 - x^{2})

limit as x approaches 0 of (ln(x))^2

x \to 0 lim ((ln (x))^{2})

limit as x approaches-1+of 1/(x^2+1)

x \to - 1 + lim (\frac{1}{x ^{2} + 1})