What is the integral of 1/(xsqrt(ln^2(x)-2)) ?

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ^{2} - 2} d u

= \int \frac{1}{u ^{2} - 2} d u

Apply Trigonometric Substitution

: \int sec (v) d v

= \int sec (v) d v

Use the common integral:

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

= ln ∣ tan (v) + sec (v) ∣

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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