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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

= \int \frac{6 u ^{3}}{1 - u} 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}}{1 - u} d u

长除

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

= 6 \cdot \int - u^{2} - u - 1 + \frac{1}{- u + 1} 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 u d u - \int 1 d u + \int \frac{1}{- u + 1} d u)

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

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

\int 1 d u = u

\int \frac{1}{- u + 1} d u = - ln ∣ - u + 1 ∣

= 6 (- \frac{u ^{3}}{3} - \frac{u ^{2}}{2} - u - ln ∣ - u + 1 ∣)

u = 6 x

代回

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

化简

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

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

解答补常数

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

作图

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