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Question Number 60240 by ANTARES VY last updated on 19/May/19

$$\underset{0}{\stackrel{2}{\int }}\frac{\mathbf{ln}\left(\mathbf{x}\right)}{\sqrt{4-{\mathbf{x}}^2}}\mathbf{dx}$$

Commented by maxmathsup by imad last updated on 19/May/19

$$letI={\int }_0^2\frac{ln\left(x\right)}{\sqrt{4-x^2}}dxchang.x=2sintgiveI={\int }_0^{\frac{\pi }{2}}\frac{ln\left(2\right)+ln\left(sint\right)}{2cost}2costdt$$$$=\frac{\pi }{2}ln\left(2\right)+{\int }_0^{\frac{\pi }{2}}ln\left(sint\right)dtbut{\int }_0^{\frac{\pi }{2}}ln\left(sint\right)dt=-\frac{\pi }{2}ln\left(2\right)\left(resultproved\right)\Rightarrow$$$$I=0.$$

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