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

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

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

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

ln \frac{x ^{2} - a}{a} + \frac{x}{a} + C

Solution steps

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

Apply Trigonometric Substitution

: \int sec (u) d u

= \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}{a} x)

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

Simplify

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

= ln \frac{x ^{2} - a}{a} + \frac{x}{a}

Add a constant to the solution

= ln \frac{x ^{2} - a}{a} + \frac{x}{a} + C

\frac{d}{d x} (e^{9 x^{2}} + x)

\frac{3 s e ^{- s}}{2 s ^{2} + 8}

x \to 2 - lim (- \frac{2}{2 - x})

- 2 y^{^{'}} + \frac{4 y}{x} = x^{4} y^{3}

\frac{\partial}{\partial x} (x^{3} - 3 x y^{2})

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