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

The integral of ln(a^2-x^2) is xln(a^2-x^2)+aln| x/a+1|-aln| x/a-1|-2x+C

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

integral

ln (a^{2} - x^{2})

x ln (a^{2} - x^{2}) + a ln \frac{x}{a} + 1 - a ln \frac{x}{a} - 1 - 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

\int - \frac{2 x ^{2}}{a ^{2} - x ^{2}} d x = - a ln \frac{x}{a} + 1 + a ln \frac{x}{a} - 1 + 2 x

= x ln (a^{2} - x^{2}) - (- a ln \frac{x}{a} + 1 + a ln \frac{x}{a} - 1 + 2 x)

Simplify

= x ln (a^{2} - x^{2}) + a ln \frac{x}{a} + 1 - a ln \frac{x}{a} - 1 - 2 x

Add a constant to the solution

= x ln (a^{2} - x^{2}) + a ln \frac{x}{a} + 1 - a ln \frac{x}{a} - 1 - 2 x + C

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