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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

= \int - \frac{1}{u + 3 u} d u

提出常数:

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

= - \int \frac{1}{u + 3 u} d u

使用换元积分法

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

提出常数:

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

= - 6 \cdot \int \frac{v ^{3}}{v + 1} d v

使用换元积分法

= - 6 \cdot \int \frac{( w - 1 ) ^{3}}{w} d w

乘开

\frac{( w - 1 ) ^{3}}{w} : w^{2} - 3 w + 3 - \frac{1}{w}

= - 6 \cdot \int w^{2} - 3 w + 3 - \frac{1}{w} d w

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 w^{2} d w - \int 3 w d w + \int 3 d w - \int \frac{1}{w} d w)

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

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

\int 3 d w = 3 w

\int \frac{1}{w} d w = ln ∣ w ∣

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

代回

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

解答补常数

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

作图

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