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Category: Integration

The-integral-0-1-2-ln-1-2x-1-4x-2-dx-a-pi-4-ln2-b-pi-8-ln2-c-pi-16-ln2-d-pi-32-ln2-

Question Number 49827 by rahul 19 last updated on 11/Dec/18 $${The}\:{integral}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }{dx}\:=\:? \\ $$$$\left.{a}\left.\right)\left.\:\left.\frac{\pi}{\mathrm{4}}{ln}\mathrm{2}\:\:\:\:{b}\right)\frac{\pi}{\mathrm{8}}{ln}\mathrm{2}\:\:\:\:{c}\right)\frac{\pi}{\mathrm{16}}{ln}\mathrm{2}\:\:\:{d}\right)\frac{\pi}{\mathrm{32}}{ln}\mathrm{2} \\ $$ Commented by rahul 19 last updated on…

find-0-cos-pix-2-x-2-3-2-dx-

Question Number 115361 by Bird last updated on 25/Sep/20 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$ Answered by Olaf last updated on 27/Sep/20 $$\frac{\pi\left[\mathrm{6}\pi\left(\boldsymbol{\mathrm{C}}\left(\sqrt{\mathrm{6}}\right)−\boldsymbol{\mathrm{S}}\left(\sqrt{\mathrm{6}}\right)\right)+\boldsymbol{\mathrm{C}}\left(\sqrt{\mathrm{6}}\right)−\boldsymbol{\mathrm{S}}\left(\sqrt{\mathrm{6}}\right)+\sqrt{\mathrm{6}}−\mathrm{1}\right]}{\mathrm{12}\sqrt{\mathrm{3}}}…

let-f-x-0-pi-4-ln-1-x-2-cos-d-with-x-lt-1-1-find-a-explicit-form-of-f-x-2-calculate-0-pi-4-ln-1-1-4-cos-d-

Question Number 49806 by maxmathsup by imad last updated on 10/Dec/18 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} {cos}\theta\right){d}\theta\:\:\:{with}\:\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}{cos}\theta\right){d}\theta\:. \\ $$ Commented by…

sin-8-x-cos-8-x-1-2sin-2-x-cos-2-x-a-1-2-sin-2x-b-1-2-sin-2x-c-None-

Question Number 49746 by rahul 19 last updated on 10/Dec/18 $$\int\frac{\mathrm{sin}^{\mathrm{8}} {x}−\mathrm{cos}^{\mathrm{8}} {x}}{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} {x}.\mathrm{cos}^{\mathrm{2}} {x}}\:=\:? \\ $$$$\left.{a}\left.\right)\left.\:\frac{−\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}\:\:\:{b}\right)\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}\:\:\:{c}\right){None}. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…