What is the integral of 1/(sqrt(x^2-12)) ?

The integral of 1/(sqrt(x^2-12)) is ln(1/(2sqrt(3))|sqrt(x^2-12)+x|)+C

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integral of 1/(sqrt(x^2-12))

\int \frac{1}{x ^{2} - 12} d x

ln (\frac{1}{2 3} x^{2} - 12 + x) + C

Solution steps

\int \frac{1}{x ^{2} - 12} d x

Apply Trigonometric Substitution

= \int sec (u) d u

Use the common integral:

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

= ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arcsec (\frac{1}{2 3} x)

= ln tan (arcsec (\frac{1}{2 3} x)) + sec (arcsec (\frac{1}{2 3} x))

Simplify

ln tan (arcsec (\frac{1}{2 3} x)) + sec (arcsec (\frac{1}{2 3} x)) : ln (\frac{1}{2 3} x^{2} - 12 + x)

= ln (\frac{1}{2 3} x^{2} - 12 + x)

Add a constant to the solution

= ln (\frac{1}{2 3} x^{2} - 12 + x) + C

What is the integral of 1/(sqrt(x^2-12)) ?
The integral of 1/(sqrt(x^2-12)) is ln(1/(2sqrt(3))|sqrt(x^2-12)+x|)+C