Solve the integral of logarithmic functions int(ln(x^2+1))dx

 Exercise

$\int\ln\left(x^2+1\right)dx$

 

The integral $\int\ln\left(x^2+1\right)dx$ results in $\left(x^2+1\right)\ln\left(x^2+1\right)-\left(x^2+1\right)$

$\left(x^2+1\right)\ln\left|x^2+1\right|-\left(x^2+1\right)$

 

Simplify the product $-(x^2+1)$

$\left(x^2+1\right)\ln\left|x^2+1\right|-x^2-1$

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\left(x^2+1\right)\ln\left|x^2+1\right|-x^2-1+C_0$

 

We can combine and rename $-1$ and $C_0$ as other constant of integration

$\left(x^2+1\right)\ln\left|x^2+1\right|-x^2+C_1$

$\left(x^2+1\right)\ln\left|x^2+1\right|-x^2+C_1$

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