Category: Integration

Question Number 2751 by prakash jain last updated on 26/Nov/15 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\:{i}}{{i}}=\frac{\pi}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by prakash jain last updated on 27/Nov/15…

Question Number 133791 by mnjuly1970 last updated on 24/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:……{advanced}\:\:\:\:{integral}…. \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}−{e}^{−\varphi{x}} }{\mathrm{1}+{e}^{\varphi{x}} }\:\right)\frac{{dx}}{{x}}\:=?? \\ $$$$\:\:\:\varphi:\:=\:{Golden}\:{ratio}… \\ $$$$ \\ $$ Answered…

Question Number 68241 by mathmax by abdo last updated on 07/Sep/19 $${calculate}\:\int\int_{{w}} \:\:\:\left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{w}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\right\} \\ $$ Commented by mathmax…

Question Number 68219 by ~ À ® @ 237 ~ last updated on 07/Sep/19 $$\:\:\:{Let}\:{consider}\:\left({a}_{{n}} \right)_{{n}} \:{and}\:\left({u}_{{n}} \right)_{{n}} \:{two}\:{reals}\:\:{sequence}\:\: \\ $$$${defined}\:{such}\:{as}\:\:\:{a}_{\mathrm{0}} =\mathrm{1}\:,\:\forall\:{n}>\mathrm{1}\:\:{a}_{{n}+\mathrm{1}} =\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}}…

Question Number 2645 by Filup last updated on 24/Nov/15 $${A}=\int_{{N}_{\mathrm{1}} } ^{{N}_{\mathrm{2}} } \lfloor{x}\rfloor{dx} \\ $$$$\left({N}_{\mathrm{1}} ,\:{N}_{\mathrm{2}} \right)\in\mathbb{Z},\:\:\:{N}_{\mathrm{1}} <{N}_{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Solve}\:\mathrm{for}\:{A} \\ $$…