What is the integral of ln(x^2+9) ?

The integral of ln(x^2+9) is xln(x^2+9)-6(-arctan(x/3)+x/3)+C

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integral of ln(x^2+9)

\int ln (x^{2} + 9) d x

x ln (x^{2} + 9) - 6 (- arctan (\frac{x}{3}) + \frac{x}{3}) + C

Solution steps

\int ln (x^{2} + 9) d x

Apply Integration By Parts

= x ln (x^{2} + 9) - \int \frac{2 x ^{2}}{x ^{2} + 9} d x

\int \frac{2 x ^{2}}{x ^{2} + 9} d x = 6 (- arctan (\frac{x}{3}) + \frac{x}{3})

= x ln (x^{2} + 9) - 6 (- arctan (\frac{x}{3}) + \frac{x}{3})

Add a constant to the solution

= x ln (x^{2} + 9) - 6 (- arctan (\frac{x}{3}) + \frac{x}{3}) + C

What is the integral of ln(x^2+9) ?
The integral of ln(x^2+9) is xln(x^2+9)-6(-arctan(x/3)+x/3)+C