Solve ∫ ln(x^2-b) wrt x

$$\int \ln ( x ^ { 2 } - b ) d x$$

Evaluate

$$x\ln(x^{2}-b)-2x-2\arctan(\sqrt{-\frac{1}{b}}x)\sqrt{-\frac{1}{b}}b+С$$

$$b<0$$

Solution Steps

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{\int \ln(x^{2}-b)\mathrm{d}x}{\ln(e)}$$

Simplify.

$$\frac{\ln(x^{2}-b)x-2b\sqrt{-\frac{1}{b}}\arctan(\sqrt{-\frac{1}{b}}x)-2x}{\ln(e)}$$

Simplify.

$$\ln(x^{2}-b)x-2b\sqrt{-\frac{1}{b}}\arctan(\sqrt{-\frac{1}{b}}x)-2x$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\begin{matrix}\ln(x^{2}-b)x-2b\sqrt{-\frac{1}{b}}\arctan(\sqrt{-\frac{1}{b}}x)-2x+С_{3},&\end{matrix}$$

Differentiate w.r.t. x

$$\ln(x^{2}-b)$$