How do you find the integral ln( x + sqrt(x^2 -1)) ?

1 Answer
Mar 14, 2018

int ln(x+sqrt(x^2-1)) dx = x ln(x+sqrt(x^2-1)) - sqrt(x^2-1) + C

Explanation:

Use a hyperbolic substitution.

Put:

x = cosh u

Then:

dx/(du) = sinh u

and:

u = ln(x+sqrt(x^2-1))

and:

int ln(x+sqrt(x^2-1)) dx = int ln(cosh u + sqrt(cosh^2 u - 1)) dx/(du) color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = int ln(cosh u + sqrt(sinh^2 u)) sinh u color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = int ln(cosh u + sinh u) sinh u color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = int ln(e^u) sinh u color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = int u sinh u color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = int (u sinh u + cosh u)-cosh u color(white)(.) du

color(white)(int ln(x+sqrt(x^2-1)) dx) = u cosh u - sinh u + C

color(white)(int ln(x+sqrt(x^2-1)) dx) = x ln(x+sqrt(x^2-1)) - sqrt(x^2-1) + C