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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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