Solve the integral of logarithmic functions int(log(1+-1*x^2))dx

 Exercise

$\int\log \left(1-x^2\right)dx$

 

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\int\frac{\ln\left(1-x^2\right)}{\ln\left(10\right)}dx$

 

Take the constant $\frac{1}{\ln\left|10\right|}$ out of the integral

$\frac{1}{\ln\left|10\right|}\int\ln\left(1-x^2\right)dx$

 

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

$\frac{1}{\ln\left|10\right|}\left(\left(-x^2+1\right)\ln\left|-x^2+1\right|-\left(-x^2+1\right)\right)$

 

Simplify the expression

$\frac{\left(-x^2+1\right)\ln\left(-x^2+1\right)+x^2-1}{\ln\left(10\right)}$

 

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

$\frac{\left(-x^2+1\right)\ln\left|-x^2+1\right|+x^2-1}{\ln\left|10\right|}+C_0$

$\frac{\left(-x^2+1\right)\ln\left|-x^2+1\right|+x^2-1}{\ln\left|10\right|}+C_0$

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