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Category: Integration

Calculus-lim-1-pi-0-2pi-k-1-n-sin-kx-2-k-2-dx-

Question Number 144053 by mnjuly1970 last updated on 21/Jun/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……..\:{Calculus}…….. \\ $$$$\:\:\:\:\:\:\:\Omega:={lim}\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{sin}\left({kx}\right)}{\:\sqrt{\mathrm{2}^{{k}} }}\right)^{\mathrm{2}} {dx}=? \\ $$$$ \\ $$ Answered…

Question-144042

Question Number 144042 by maryxxxx last updated on 20/Jun/21 Answered by mindispower last updated on 20/Jun/21 $$\int\frac{{dx}}{{x}^{\mathrm{3}} \sqrt{\mathrm{1}−\left(\frac{\mathrm{10}}{{x}}\right)^{\mathrm{2}} }} \\ $$$${y}=\frac{\mathrm{100}}{{x}^{\mathrm{2}} },{y}<\mathrm{1}\Rightarrow{dy}=\frac{−\mathrm{200}}{{x}^{\mathrm{3}} }{dx} \\ $$$$\Rightarrow\frac{−\mathrm{1}}{\mathrm{200}}\int\frac{{dy}}{\:\sqrt{\mathrm{1}−{y}}}{dy}=\frac{\mathrm{1}}{\mathrm{100}}\sqrt{\mathrm{1}−{y}}+{c}=\frac{\sqrt{\mathrm{1}−\frac{\mathrm{100}}{{x}^{\mathrm{2}}…

prove-that-m-N-a-k-b-k-R-cos-2m-x-k-1-m-a-k-cos-2kx-cos-2m-1-x-k-1-m-b-k-cos-2k-1-x-and-find-expr-of-a-k-b-k-in-terms-of-k-

Question Number 144000 by bluberry508 last updated on 20/Jun/21 $$\mathrm{prove}\:\mathrm{that}\: \\ $$$$ \\ $$$$\forall_{{m}} \in\mathbb{N}\:,\:{a}_{{k}} ,{b}_{{k}} \in\mathbb{R} \\ $$$$\mathrm{cos}\:^{\mathrm{2}{m}} {x}\:=\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}{a}_{{k}} \mathrm{cos}\:\mathrm{2}{kx} \\ $$$$\mathrm{cos}\:^{\mathrm{2}{m}−\mathrm{1}}…

Question-12908

Question Number 12908 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/May/17 Answered by 433 last updated on 07/May/17 $$\int_{\sqrt{\mathrm{2}}} ^{\mathrm{2}} \left(\frac{{x}^{\mathrm{2}} }{{x}+\left[{x}+\mathrm{1}\right]}\right){dx}+\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} \left(\frac{{x}^{\mathrm{2}} }{{x}+\left[{x}+\mathrm{1}\right]}\right){dx}= \\ $$$$\int_{\sqrt{\mathrm{2}}}…

Question-143952

Question Number 143952 by akolade last updated on 19/Jun/21 Answered by Olaf_Thorendsen last updated on 20/Jun/21 $$\mathrm{C}\:=\:\int\frac{\mathrm{cosh}{x}}{\mathrm{cosh}{x}+\mathrm{sinh}{x}}\:{dx} \\ $$$$\mathrm{S}\:=\:\int\frac{\mathrm{sinh}{x}}{\mathrm{cosh}{x}+\mathrm{sinh}{x}}\:{dx} \\ $$$$\mathrm{C}+\mathrm{S}\:=\:\int{dx}\:=\:{x}+\mathrm{cst}\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{C}−\mathrm{S}\:=\:\int\frac{\mathrm{cosh}{x}−\mathrm{sinh}{x}}{\mathrm{cosh}{x}+\mathrm{sinh}{x}}\:{dx} \\ $$$$\mathrm{C}−\mathrm{S}\:=\:\int\frac{\left(\mathrm{cosh}{x}−\mathrm{sinh}{x}\right)^{\mathrm{2}}…

Question-143932

Question Number 143932 by Ar Brandon last updated on 19/Jun/21 Answered by mathmax by abdo last updated on 19/Jun/21 $$\left.\mathrm{1}\right)\:\mathrm{I}=\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} } \mathrm{dt}\:\:\:\:\:\:\Rightarrow\mathrm{I}=\int_{\mathrm{0}} ^{\mathrm{a}}…