Category: Differentiation

Question Number 79964 by ubaydulla last updated on 29/Jan/20 $$\int (\tan {}^{2}x+\tan {}^{4}x)dx=[\tan {}^{2}x=t]=\int (t^2+t^4)dx=\frac{t^3}{3}+\frac{t^5}{5}+c$$ Commented by $@ty@m123 last updated on…

Question Number 145450 by solihin last updated on 05/Jul/21 Answered by Olaf_Thorendsen last updated on 05/Jul/21 $$a)$$$$a_n=\frac{2^n}{3^{n+1}},n\in {\mathbb{N}}^{\ast }$$$${a}_{{n}} \:=\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{n}}…

Question Number 79689 by ~blr237~ last updated on 27/Jan/20 $$Find\underset{x\to \mathrm{\infty }}{\lim }{x}^{2}ln(1+\frac{1}{x})-xwithoutDLandifyouhavetousethe$$$$hospitalrulepleasejustifythatyourfunctionis{C}^1$$ Commented by mr W last updated on 27/Jan/20…

Question Number 79644 by loveineq. last updated on 26/Jan/20 $$f\left(x\right)=\frac{4(4x^2+3)}{4x^2+4x+5}$$$$\mathrm{prove}\mathrm{that}f\left(x\right)⩾2.$$ Commented by john santu last updated on 27/Jan/20 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)<\mathrm{2}…

Question Number 145163 by mim24 last updated on 02/Jul/21 Answered by mathmax by abdo last updated on 02/Jul/21 $$\mathrm{y}=\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\dots +\mathrm{\infty }}}}\Rightarrow \mathrm{y}=\sqrt{\mathrm{x}+\mathrm{y}}\Rightarrow {\mathrm{y}}^2=\mathrm{x}+\mathrm{y}\Rightarrow$$$${\mathrm{y}}^2-\mathrm{y}-\mathrm{x}=0\to \mathrm{\Delta }=1+4\mathrm{x}\Rightarrow \mathrm{y}=\frac{1+\sqrt{1+4\mathrm{x}}}{2}\mathrm{or}\mathrm{y}=\frac{1-\sqrt{1+4\mathrm{x}}}{2}$$$$\mathrm{but}\:\mathrm{y}\geqslant\mathrm{0}\:\Rightarrow\mathrm{y}=\frac{\mathrm{1}+\sqrt{\mathrm{4x}+\mathrm{1}}}{\mathrm{2}}\:\Rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{4}}{\mathrm{2}\sqrt{\mathrm{4x}+\mathrm{1}}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{4x}+\mathrm{1}}}\:\Rightarrow…