Category: Differentiation

Question Number 160159 by mnjuly1970 last updated on 25/Nov/21 $$$$$$provethat..$$$$$$$${\int }_0^{\mathrm{\infty }}\frac{x}{{cosh}^3\left(x\right)}dx=\mathrm{G}-\frac{1}{2}\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◼"\; src="https://s.w.org/images/core/emoji/15.1.0/svg/25fc.svg">}$$$$\mathrm{G}:catalanconstant$$$$$$…

Question Number 29037 by abdo imad last updated on 03/Feb/18 $$letgivethesequence\left(y_n\right)/y_0\left(x\right)=1and$$$$y_n\left(x\right)=1+{\int }_0^x{\left(y_{n-1}\left(t\right)\right)}^{2}dt,letsupposex\in [0,1]prove$$$${that}\:\left({y}_{{n}} \right)\:{is}\:{increasing}\:{majored}\:{by}\:\frac{\mathrm{1}}{\mathrm{1}−{x}}\:{if}\:{y}={lim}_{{n}\rightarrow+\infty} {y}_{{n}} \…

Question Number 28949 by amit96 last updated on 02/Feb/18 $$xy\frac{x^4-y^4}{x^4+y^4}and0fororigin$$$$thenfuntionis$$$$1.continuous$$$$2.mixpartialarenotequalatorigin$$$$3.limitatoriginis1$$ Terms…

Question Number 28903 by amit96 last updated on 01/Feb/18 Commented by abdo imad last updated on 01/Feb/18 $$letputI={\int }_{\frac{1}{2}}^1\left[\frac{1}{x^2}\right]dxandusethech.\frac{1}{x^2}=t\iff x^2=\frac{1}{t}$$$$\Leftrightarrow{x}=\frac{\mathrm{1}}{\:\sqrt{{t}}}\:\Rightarrow\:{dx}=\frac{−\mathrm{1}}{\mathrm{2}{t}\sqrt{{t}}}{dt}\:\:\:{and}\:\:{I}=\:−\int_{\mathrm{1}}…

Question Number 159966 by malwaan last updated on 23/Nov/21 $$ay=\sqrt{x+\sqrt{x+\sqrt{x+\dots ..}}}$$$$by=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\dots ..}}}}$$$$find\frac{dy}{dx}$$ Commented by tounghoungko last updated on 23/Nov/21 $$\left({a}\right){y}=\sqrt{{x}+{y}} \…