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

解答

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

ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

求解步骤

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

使用积分常见定则:

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

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

化简

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

= ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣

解答补常数

= ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

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