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受欢迎的微积分 >

integral of (sqrt(w))/(sqrt(1-\sqrt{w))}

解答

\int \frac{w}{1 - w} d w

解答

- 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w) + C

求解步骤

\int \frac{w}{1 - w} d w

使用换元积分法

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

提出常数:

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

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

使用换元积分法

= 2 \cdot \int - 2 (- v^{2} + 1)^{2} d v

提出常数:

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

= 2 (- 2 \cdot \int (- v^{2} + 1)^{2} d v)

乘开

(- v^{2} + 1)^{2} : v^{4} - 2 v^{2} + 1

= 2 (- 2 \cdot \int v^{4} - 2 v^{2} + 1 d v)

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

= 2 (- 2 (\int v^{4} d v - \int 2 v^{2} d v + \int 1 d v))

\int v^{4} d v = \frac{v ^{5}}{5}

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

\int 1 d v = v

= 2 (- 2 (\frac{v ^{5}}{5} - \frac{2 v ^{3}}{3} + v))

代回

= 2 - 2 \frac{( 1 - w ) ^{5}}{5} - \frac{2 ( 1 - w ) ^{3}}{3} + 1 - w

化简

2 - 2 \frac{( 1 - w ) ^{5}}{5} - \frac{2 ( 1 - w ) ^{3}}{3} + 1 - w : - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w)

= - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w)

解答补常数

= - 4 (\frac{1}{5} (1 - w)^{\frac{5}{2}} - \frac{2}{3} (1 - w)^{\frac{3}{2}} + 1 - w) + C

作图

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