What is the integral of ln|x^2+1| ?

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

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integral of ln|x^2+1|

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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