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

The integral of ln(9-x^2) is xln(9-x^2)+6(1/2 ln| x/3+1|-1/2 ln| x/3-1|-x/3)+C

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integral of ln(9-x^2)

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

x ln (9 - x^{2}) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2 x ^{2}}{9 - x ^{2}} d x = - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

= x ln (9 - x^{2}) - (- 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}))

Simplify

= x ln (9 - x^{2}) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

Add a constant to the solution

= x ln (9 - x^{2}) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + C

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What is the integral of ln(9-x^2) ?
The integral of ln(9-x^2) is xln(9-x^2)+6(1/2 ln| x/3+1|-1/2 ln| x/3-1|-x/3)+C