find-the-integral-displaystyle-int-fracsqrt9-ln-x2xln-xdx

Question

Find the integral: ∫9−(ln⁡x)2x ln⁡x dx\displaystyle \int \frac{\sqrt{9-(\ln $$x)^2$$}}{x\,\ln x}\,dx

Accepted Answer

−3ln⁡∣3+9−(ln⁡x)2ln⁡x∣+9−(ln⁡x)2+C{\displaystyle-3\ln\left|\frac{3+\sqrt{9-(\ln $$x)^2$$}}{\ln x}\right|+\sqrt{9-(\ln $$x)^2$$}+C}

Answer

−3ln⁡∣3+9−(ln⁡x)2ln⁡x∣−9−(ln⁡x)2+C{{{{\displaystyle-3\ln\left|\frac{3+\sqrt{9-(\ln $$x)^2$$}}{\ln x}\right|-\sqrt{9-(\ln $$x)^2$$}+C}}}}

Answer

−3ln⁡(3+9−(ln⁡x)2)+9−(ln⁡x)2+C{{\displaystyle-3\ln\left(3+\sqrt{9-(\ln $$x)^2$$}\right)+\sqrt{9-(\ln $$x)^2$$}+C}}

Answer

3ln⁡(3+9−(ln⁡x)2)+9−(ln⁡x)2+C{{{\displaystyle3\ln\left(3+\sqrt{9-(\ln $$x)^2$$}\right)+\sqrt{9-(\ln $$x)^2$$}+C}}}

Find the integral: ∫9−(lnx)2xlnxdx\(\displaystyle\) \(\int\) \(\frac{\sqrt{9-(\ln x)^2}\)}{x\,\(\ln\) x}\,dx