What is the integral of ln(x+sqrt(1+x^2)) ?

The integral of ln(x+sqrt(1+x^2)) is xln(x+sqrt(1+x^2))-sqrt(1+x^2)+C

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integral of ln(x+sqrt(1+x^2))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} ((3 x + 4)^{2})

x \to - 2 lim (- 2)

\int \frac{3 y}{( 2 + 3 y ^{2} ) ^{3}} d y

\frac{d}{d x} (cos^{2} (3 x + 1) - sin^{2} (3 x + 1))

\int 1 + sin (x) d x

What is the integral of ln(x+sqrt(1+x^2)) ?
The integral of ln(x+sqrt(1+x^2)) is xln(x+sqrt(1+x^2))-sqrt(1+x^2)+C