Does $\int^3_0 (9-x^2)^{-3/2}\, dx$ converge?

Question

Does $$\int^3_0 {dx \over (9-x^2)^{3 \over 2}}$$ converge?

I tried to compare to $1 \over x^3$ and $1 \over x^{3 \over 2}$ using compare test and limit compare test but it didn't work out. (I don't want to calculate directly $lim_{\epsilon \to 3^+}\int^{3- \epsilon}_0 {dx \over (9-x^2)^{3 \over 2}}$ and to see if this limit exists or not)

Answer 1

Note that $9-x^2=(3+x)(3-x) $. So your integral is comparable to $$ \int_0^3\frac1 {(3-x)^{3/2}}\,dx. $$