What is the integral of 1/(sqrt(x^2-25)) ?

The integral of 1/(sqrt(x^2-25)) is ln(1/5 |sqrt(x^2-25)+x|)+C

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integral of 1/(sqrt(x^2-25))

\int \frac{1}{x ^{2} - 25} d x

ln (\frac{1}{5} x^{2} - 25 + x) + C

Solution steps

\int \frac{1}{x ^{2} - 25} 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}{5} x)

= ln tan (arcsec (\frac{1}{5} x)) + sec (arcsec (\frac{1}{5} x))

Simplify

ln tan (arcsec (\frac{1}{5} x)) + sec (arcsec (\frac{1}{5} x)) : ln (\frac{1}{5} x^{2} - 25 + x)

= ln (\frac{1}{5} x^{2} - 25 + x)

Add a constant to the solution

= ln (\frac{1}{5} x^{2} - 25 + x) + C

\int (\frac{3}{x ^{2}}) d x

\int \frac{2}{3} x^{2} ln (x) d x

x \to 1 lim (\frac{( x ^{2} - 5 )}{x})

x \to 5 lim (8 x + 5)

\frac{d}{d x} (e^{x^{2} y} - 2 x - 2 y)

What is the integral of 1/(sqrt(x^2-25)) ?
The integral of 1/(sqrt(x^2-25)) is ln(1/5 |sqrt(x^2-25)+x|)+C