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

解答

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

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

求解步骤

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

使用三角换元法

= 2 \cdot \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 2 ln ∣ tan (u) + sec (u) ∣

u = arcsec (\frac{1}{2} x)

代回

= 2 ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x))

化简

2 ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x)) : 2 ln \frac{1}{2} (x^{2} - 2) + \frac{1}{2} x

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

解答补常数

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

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