What is the integral of ln(x^2+a^2) ?

The integral of ln(x^2+a^2) is xln(x^2+a^2)+2aarctan(x/a)-2x+C

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integral of ln(x^2+a^2)

\int ln (x^{2} + a^{2}) d x

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

Solution steps

\int ln (x^{2} + a^{2}) d x

Apply Integration By Parts

: x ln (x^{2} + a^{2}) - \int \frac{2 x ^{2}}{x ^{2} + a ^{2}} d x

= x ln (x^{2} + a^{2}) - \int \frac{2 x ^{2}}{x ^{2} + a ^{2}} d x

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

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

Simplify

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

Add a constant to the solution

= x ln (x^{2} + a^{2}) + 2 a arctan (\frac{x}{a}) - 2 x + C

\frac{\partial}{\partial x} (x + i y)

\int \frac{( x - 1 )}{x ^{3} - x ^{2} - 2 x} d x

2 - x

\int (x + 1) sin (x^{2} + 2 x) d x

x \to 1 lim (sin (5 π x))

What is the integral of ln(x^2+a^2) ?
The integral of ln(x^2+a^2) is xln(x^2+a^2)+2aarctan(x/a)-2x+C