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

\int \frac{121 - x ^{2}}{x} d x

\frac{d}{d x} (\frac{1}{x \cdot x})

\int 2 t^{3} d t

\frac{t}{( t - 2 ) ^{2}}

5 y^{^{''}} - 4 y^{^{'}} = 0

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