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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

y^{^{''}} + 8 y^{^{'}} = 0

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

f (x) = 3 - \frac{18}{5 x ^{\frac{1}{10}}}

\int \frac{1}{x ^{2} - 2 x + 17} d x

\frac{3 - 2 s}{s ^{2} + 4 s + 5}

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