Category: Differentiation

Question Number 5513 by 314159 last updated on 17/May/16 $$Ifn>1,provebymathematicalinductionthat$$$$n({(n+1)}^{\frac{1}{n}}-1)<1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots \frac{1}{n}.$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 136570 by mnjuly1970 last updated on 23/Mar/21 $$\dots \dots .nice\dots ..calculus\dots ..$$$$\mathrm{\Omega }=\underset{n=0}{\sum ^{\mathrm{\infty }}}{cos}^n\left(x\right).cos\left(nx\right)=?$$$$solution::::$$$$\mathrm{\Omega }=\frac{1}{2}\underset{n=0}{\sum ^{\mathrm{\infty }}}{cos}^{n-1}\left(x\right)\{cos(x-nx)+cos(x+nx)$$$$\:\:\therefore\:\mathrm{2}\Omega=\underset{{n}=\mathrm{0}} {\overset{\infty}…

Question Number 5456 by 3 last updated on 15/May/16 $$⪹$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 136295 by mnjuly1970 last updated on 20/Mar/21 $$\dots .nicecalculus\dots$$$$prove::$$$$1::\mathit{\varphi }={\int }_0^{\frac{\pi }{2}}\frac{{tan}^{-1}\left(\sqrt{tan\left(x\right)}\right)}{tan\left(x\right)}dx=\frac{\pi }{2}log(2+\sqrt{2})$$$$\mathrm{2}::\Omega=\int_{−\pi} ^{\:\pi} \frac{{e}^{\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)} {cos}\left({sin}\left({x}\right)\right)}{{e}^{{x}} +{e}^{{sin}\left({x}\right)} }{dx}=\pi \…

Question Number 70741 by aliesam last updated on 07/Oct/19 $$f\left(x\right)=\{\begin{array}{l}{x}^{2}x⩽1\\ \\ \mid x-2\mid x>1\end{array}$$$$$$$$findthecriticalpoints$$ Commented by kaivan.ahmadi last updated on 07/Oct/19 $${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}}…