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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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