integral of 1/(sqrt(x^2-36))

解答

\int \frac{1}{x ^{2} - 36} d x

ln (\frac{1}{6} x^{2} - 36 + x) + C

求解步骤

\int \frac{1}{x ^{2} - 36} d x

使用三角换元法

= \int sec (u) d u

使用积分常见定则:

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

= ln ∣ tan (u) + sec (u) ∣

u = arcsec (\frac{1}{6} x)

代回

= ln tan (arcsec (\frac{1}{6} x)) + sec (arcsec (\frac{1}{6} x))

化简

ln tan (arcsec (\frac{1}{6} x)) + sec (arcsec (\frac{1}{6} x)) : ln (\frac{1}{6} x^{2} - 36 + x)

= ln (\frac{1}{6} x^{2} - 36 + x)

解答补常数

= ln (\frac{1}{6} x^{2} - 36 + x) + C

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