What is the integral of ln(x+sqrt((x^2-1))) ?

The integral of ln(x+sqrt((x^2-1))) is xln(x+sqrt(x^2-1))-sqrt(x^2-1)+C

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integral of ln(x+sqrt((x^2-1)))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

ln (sec (x) + tan (x))

\int 1 + 9 sin^{2} (3 x) d x

x \to 0 lim (\frac{( x ^{2} - 4 x )}{x})

\int - e^{x} (x + 1) d x

y = \frac{x ^{2} - 4}{x}

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