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Category: Differentiation

nice-calculus-prove-that-0-ln-1-x-2-1-x-2-2-dx-pi-2-ln-2-pi-4-

Question Number 136555 by mnjuly1970 last updated on 23/Mar/21 $$\:\:\:\:\:\:\:\:\:….{nice}\:\:\:\:……\:\:\:{calculus}…. \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}}{ln}\left(\mathrm{2}\right)−\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\::::::::: \\ $$ Answered by…

nice-calculus-prove-1-0-pi-2-tan-1-tan-x-tan-x-dx-pi-2-log-2-2-2-pi-pi-e-sin-x-cos-x-cos-sin-x-e-x-e-sin-x-dx-pi-

Question Number 136295 by mnjuly1970 last updated on 20/Mar/21 $$\:\:\:\:\:\:….{nice}\:\:{calculus}… \\ $$$${prove}\::: \\ $$$$\mathrm{1}\:::\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{tan}^{−\mathrm{1}} \left(\sqrt{{tan}\left({x}\right)}\:\right)}{{tan}\left({x}\right)}{dx}=\frac{\pi}{\mathrm{2}}{log}\left(\mathrm{2}+\sqrt{\mathrm{2}}\:\right) \\ $$$$\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 \\…

f-x-x-2-x-1-x-2-x-gt-1-find-the-critical-points-

Question Number 70741 by aliesam last updated on 07/Oct/19 $${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\leqslant\mathrm{1}}\\{}\\{\mid{x}−\mathrm{2}\mid\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}>\mathrm{1}}\end{cases} \\ $$$$ \\ $$$${find}\:{the}\:{critical}\:{points} \\ $$ Commented by kaivan.ahmadi last updated on 07/Oct/19 $${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}}…

Can-you-show-why-a-d-dx-sin-x-cos-x-b-d-dx-cos-x-sin-x-

Question Number 5171 by FilupSmith last updated on 25/Apr/16 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{show}\:\mathrm{why}: \\ $$$$\left({a}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{sin}\:{x}\right)=\mathrm{cos}\:{x} \\ $$$$\left({b}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{cos}\:{x}\right)=−\mathrm{sin}\:{x} \\ $$ Commented by FilupSmith last updated on 25/Apr/16 $${f}\:'\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}}…

d-dx-log-10-x-

Question Number 70705 by sadimuhmud 136 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{log}_{\mathrm{10}} \mathrm{x}\right)=? \\ $$ Answered by $@ty@m123 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{log}_{{e}} \:{x}.}{\mathrm{log}_{{e}} \:\mathrm{10}}\right)=\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{10}}.\frac{{d}}{{dx}}\left(\mathrm{ln}\:{x}\right) \\…