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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x arctan (y))

n = 3 \sum \infty (\frac{9}{11})^{- n}

\int cos^{5} (4 - x) sin (4 - x) d x

\int \frac{x ^{2} - 3 x + x}{x ^{2}} d x

\frac{d}{d x} (- \frac{6}{( x - 1 ) ^{4}})

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