Category: Integration

Question Number 2435 by Yozzi last updated on 20/Nov/15 $$Provethat$$$${\mathrm{\Gamma }}^{\prime }\left(1\right)={\int }_0^{\mathrm{\infty }}{e}^{-x}lnxdx$$$$isanegativenumber.$$ Commented by prakash jain last updated…

Question Number 67963 by mhmd last updated on 02/Sep/19 Commented by mathmax by abdo last updated on 02/Sep/19 $${generally}\:{if}\:{F}\left({x}\right)=\int_{{u}\left({x}\right)} ^{{v}\left({x}\right)} {f}\left({t}\right){dt}\:\Rightarrow{F}^{'} \left({x}\right)={v}^{'} \left({x}\right){f}\left({v}\left({x}\right)\right)−{u}^{'} \left({x}\right){f}\left({u}\left({x}\right)\right) \…

Question Number 133490 by mnjuly1970 last updated on 22/Feb/21 $$\dots .advncedcalculus\dots .$$$$prove:\mathit{\varphi }={\int }_0^{\mathrm{\infty }}\frac{sin\left(tan\left(x\right)\right)}{x}dx=\frac{\pi }{2}(1-\frac{1}{e})$$$$provethat:\mathbf{\Phi }={\int }_0^{\mathrm{\infty }}\frac{1-e^{-x^2}}{x^2}dx=\sqrt{\pi }$$$$$$$$\:\:\:…

Question Number 67937 by A8;15: last updated on 02/Sep/19 Commented by mathmax by abdo last updated on 02/Sep/19 $$atformofserie$$$${I}\:=\int\sqrt{{e}^{{x}} }{dx}\:=\:\int\:\:{e}^{\frac{{x}}{\mathrm{2}}} {dx}\:=\int\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(\frac{{x}}{\mathrm{2}}\right)^{{n}}…

Question Number 67932 by mathmax by abdo last updated on 02/Sep/19 $$letA\left(\theta \right)={\int }_0^{\mathrm{\infty }}\frac{dx}{(x^2+3)(x^4-e^{i\theta })}with0<\theta <\frac{\pi }{2}$$$$1)calculateA\left(\theta \right)intermsof\theta$$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta}…