$\int_0^4\frac{\log x}{\sqrt{4x-x^2}} dx=0$

$$4x-x^2=2^2-(x-2)^2$$

Let $x-2=2\cos2t$

$$\int_0^4\dfrac{\ln(x)}{\sqrt{4x-x^2}}dx=-4\int_{\pi/2}^0\ln(2+2\cos2t)dt$$

$$=2\ln2\int_0^{\pi/2}dt+4\int_0^{\pi/2}\ln(\cos t)dt$$

See Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$

Tags:

Calculus

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