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

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

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integral of ln(x+sqrt(x^2+a^2))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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