Category: Integration

Question Number 135103 by frc2crc last updated on 10/Mar/21 $$f\left(x\right)=1+\underset{n=2}{\sum ^{\mathrm{\infty }}}\frac{{(-x)}^n}{n}$$ Commented by Dwaipayan Shikari last updated on 10/Mar/21 $$\mathrm{1}−\left(−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{3}}…

Question Number 69564 by mathmax by abdo last updated on 25/Sep/19 $$letf\left(a\right)={\int }_0^{\mathrm{\infty }}\frac{dx}{x^4-2x^2+a}witharealanda>1$$$$1)determineaexplicitformforf\left(a\right)$$$$\left.\mathrm{2}\right)\:\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} }…

Question Number 3988 by Filup last updated on 26/Dec/15 $$S={\int }_{-\mathrm{\infty }}^0\left[\sin \left(e^x\right)\cos \left(e^{-x}\right)\right]dx$$$$$$$$\left(1\right)\mathrm{Solve}S$$$$$$$$\mathrm{Also},\mathrm{Does}:$$$$\left(\mathrm{2}\right)\:\:\:\int_{\mathrm{0}} ^{\:\infty}…

Question Number 69502 by 20190927 last updated on 24/Sep/19 $$\int \frac{3\mathrm{sinx}+4\mathrm{cosx}}{4\mathrm{sinx}+3\mathrm{cosx}}\mathrm{dx}$$ Commented by mathmax by abdo last updated on 24/Sep/19 $$letI=\int \frac{3sinx+4cosx}{4sinx+3cosx}dxchangementtan\left(\frac{x}{2}\right)=tgive$$$${I}\:=\int\frac{\mathrm{3}\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }+\mathrm{4}\frac{\mathrm{1}−{t}^{\mathrm{2}}…

Question Number 134962 by 0731619177 last updated on 09/Mar/21 Answered by Olaf last updated on 09/Mar/21 $$\mathrm{\forall }x\in {\mathbb{R}}^{\ast },\arctan x+\arctan \frac{1}{x}=\frac{\pi }{2}$$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{x}\mathrm{arctan}{x}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \…