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受欢迎的微积分 >

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

解答

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

解答

6 (\frac{1}{3} x + \frac{3}{2} 3 x + 9 6 x + 27 ln 6 x - 3) + C

求解步骤

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

使用换元积分法

= \int \frac{6 u ^{3}}{u - 3} d u

提出常数:

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

= 6 \cdot \int \frac{u ^{3}}{u - 3} d u

长除

\frac{u ^{3}}{u - 3} : u^{2} + 3 u + 9 + \frac{27}{u - 3}

= 6 \cdot \int u^{2} + 3 u + 9 + \frac{27}{u - 3} 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

= 6 (\int u^{2} d u + \int 3 u d u + \int 9 d u + \int \frac{27}{u - 3} d u)

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

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

\int 9 d u = 9 u

\int \frac{27}{u - 3} d u = 27 ln ∣ u - 3 ∣

= 6 (\frac{u ^{3}}{3} + \frac{3 u ^{2}}{2} + 9 u + 27 ln ∣ u - 3 ∣)

u = 6 x

代回

= 6 (\frac{( 6 x ) ^{3}}{3} + \frac{3 ( 6 x ) ^{2}}{2} + 9 6 x + 27 ln 6 x - 3)

化简

6 (\frac{( 6 x ) ^{3}}{3} + \frac{3 ( 6 x ) ^{2}}{2} + 9 6 x + 27 ln 6 x - 3) : 6 (\frac{1}{3} x + \frac{3}{2} 3 x + 9 6 x + 27 ln 6 x - 3)

= 6 (\frac{1}{3} x + \frac{3}{2} 3 x + 9 6 x + 27 ln 6 x - 3)

解答补常数

= 6 (\frac{1}{3} x + \frac{3}{2} 3 x + 9 6 x + 27 ln 6 x - 3) + C

作图

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