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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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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(1+x^2)+C