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受欢迎的微积分 >

integral of 1/(x^2)sqrt((1-x)/(1+x))

解答

\int \frac{1}{x ^{2}} \frac{1 - x}{1 + x} d x

解答

ln \frac{1 - x}{1 + x} + 1 + \frac{1}{\frac{1 - x}{1 + x} + 1} - ln \frac{1 - x}{1 + x} - 1 + \frac{1}{\frac{1 - x}{1 + x} - 1} + C

求解步骤

\int \frac{1}{x ^{2}} \frac{1 - x}{1 + x} d x

使用换元积分法

= \int - \frac{4 u ^{2}}{( u ^{2} - 1 ) ^{2}} d u

提出常数:

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

= - 4 \cdot \int \frac{u ^{2}}{( u ^{2} - 1 ) ^{2}} d u

将

\frac{u ^{2}}{( u ^{2} - 1 ) ^{2}}

用部份分式展开:

- \frac{1}{4 ( u + 1 )} + \frac{1}{4 ( u + 1 ) ^{2}} + \frac{1}{4 ( u - 1 )} + \frac{1}{4 ( u - 1 ) ^{2}}

= - 4 \cdot \int - \frac{1}{4 ( u + 1 )} + \frac{1}{4 ( u + 1 ) ^{2}} + \frac{1}{4 ( u - 1 )} + \frac{1}{4 ( u - 1 ) ^{2}} 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

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

\int \frac{1}{4 ( u + 1 )} d u = \frac{1}{4} ln ∣ u + 1 ∣

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

\int \frac{1}{4 ( u - 1 )} d u = \frac{1}{4} ln ∣ u - 1 ∣

\int \frac{1}{4 ( u - 1 ) ^{2}} d u = - \frac{1}{4 ( u - 1 )}

= - 4 (- \frac{1}{4} ln ∣ u + 1 ∣ - \frac{1}{4 ( u + 1 )} + \frac{1}{4} ln ∣ u - 1 ∣ - \frac{1}{4 ( u - 1 )})

u = \frac{1 - x}{1 + x}

代回

= - 4 - \frac{1}{4} ln \frac{1 - x}{1 + x} + 1 - \frac{1}{4 ( \frac{1 - x}{1 + x} + 1 )} + \frac{1}{4} ln \frac{1 - x}{1 + x} - 1 - \frac{1}{4 ( \frac{1 - x}{1 + x} - 1 )}

化简

- 4 - \frac{1}{4} ln \frac{1 - x}{1 + x} + 1 - \frac{1}{4 ( \frac{1 - x}{1 + x} + 1 )} + \frac{1}{4} ln \frac{1 - x}{1 + x} - 1 - \frac{1}{4 ( \frac{1 - x}{1 + x} - 1 )} : ln \frac{1 - x}{1 + x} + 1 + \frac{1}{\frac{1 - x}{1 + x} + 1} - ln \frac{1 - x}{1 + x} - 1 + \frac{1}{\frac{1 - x}{1 + x} - 1}

= ln \frac{1 - x}{1 + x} + 1 + \frac{1}{\frac{1 - x}{1 + x} + 1} - ln \frac{1 - x}{1 + x} - 1 + \frac{1}{\frac{1 - x}{1 + x} - 1}

解答补常数

= ln \frac{1 - x}{1 + x} + 1 + \frac{1}{\frac{1 - x}{1 + x} + 1} - ln \frac{1 - x}{1 + x} - 1 + \frac{1}{\frac{1 - x}{1 + x} - 1} + C

作图

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