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

\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

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