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Question Number 121862 by Bird last updated on 12/Nov/20

$$1)explicitef\left(a\right)={\int }_0^{\mathrm{\infty }}\frac{t^{a-1}lnt}{1+t}dt$$ $$with0<a<1$$ $$2)calculate{\int }_0^{\mathrm{\infty }}\frac{lnt}{(1+t)\sqrt{t}}dt$$

Answered by mnjuly1970 last updated on 12/Nov/20

$$solution:1::g\left(b\right)={\int }_0^{\mathrm{\infty }}\frac{t^{a+b-1}}{1+t}dt$$ $$f\left(a\right)=g^{\prime }\left(0\right)$$ $$g\left(b\right)=\mathrm{\Gamma }(a+b)\mathrm{\Gamma }(1-a-b)=\frac{\pi }{sin\left(\pi (a+b)\right)}$$ $$g^{\prime }\left(b\right)=\frac{-{\pi }^{2}cos\left(\pi (a+b)\right)}{{sin}^2\left(\pi (a+b)\right)}$$ $$\text{Missing left or extra right}$$ $$f\left(a\right)=-{\pi }^{2}cot\left(\pi a\right)csc\left(\pi a\right)...$$ $$2::f\left(\frac{1}{2}\right)=0$$ $$weknowthat::$$ $${\int }_0^{\mathrm{\infty }}\frac{ln\left(x\right)}{1+x^2}dx\stackrel{easy}{=}0$$ $$$$

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