Solve ∫ log_{e}(2a+x^2)

$$\int \log _ { e } ( 2 a + x ^ { 2 } )$$

Evaluate

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

$$a>0$$

Solution Steps

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

$$\frac{\int \ln(2a+x^{2})\mathrm{d}x}{\ln(e)}$$

Simplify.

$$\frac{\ln(2a+x^{2})x+2\sqrt{a}\sqrt{2}\arctan(\frac{\sqrt{2}x}{2\sqrt{a}})-2x}{\ln(e)}$$

Simplify.

$$\ln(2a+x^{2})x+2\sqrt{a}\sqrt{2}\arctan(\frac{\sqrt{2}x}{2\sqrt{a}})-2x$$

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.

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

Differentiate w.r.t. x

$$\ln(x^{2}+2a)$$