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

解答

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

- ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 2 x + C

求解步骤

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

提出常数:

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

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

因式分解

x^{2} - 1 : - (- x^{2} + 1)

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

提出常数:

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

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

\frac{x ^{2}}{- x ^{2} + 1} = \frac{1}{- x ^{2} + 1} - 1

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

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

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

\int 1 d x = x

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

化简

2 (- (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2} - x)) : - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 2 x

= - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 2 x

解答补常数

= - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 2 x + C

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