What is the integral of (-1)/(xsqrt(x^2-4)) ?

The integral of (-1)/(xsqrt(x^2-4)) is -1/2 arcsec(1/2 x)+C

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integral of (-1)/(xsqrt(x^2-4))

\int \frac{- 1}{x x ^{2} - 4} d x

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

Solution steps

\int \frac{- 1}{x x ^{2} - 4} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{x x ^{2} - 4} d x

Apply Trigonometric Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Integral of a constant:

\int a d x = a x

= - \frac{1}{2} \cdot 1 \cdot u

Substitute back

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

= - \frac{1}{2} \cdot 1 \cdot arcsec (\frac{1}{2} x)

Simplify

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

Add a constant to the solution

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

What is the integral of (-1)/(xsqrt(x^2-4)) ?
The integral of (-1)/(xsqrt(x^2-4)) is -1/2 arcsec(1/2 x)+C