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Category: Integration

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…

If-f-x-is-a-differentiable-function-defined-x-R-such-that-f-x-3-x-f-x-0-then-0-2-f-1-x-dx-

Question Number 115267 by bobhans last updated on 24/Sep/20 $${If}\:{f}\left({x}\right)\:{is}\:{a}\:{differentiable}\:{function} \\ $$$${defined}\:\:\forall{x}\in\mathbb{R}\:{such}\:{that}\:\left({f}\left({x}\right)\right)^{\mathrm{3}} −{x}+{f}\left({x}\right)=\mathrm{0} \\ $$$${then}\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{2}}} {\int}}\:{f}^{−\mathrm{1}} \left({x}\right)\:{dx}\:=\: \\ $$ Answered by Olaf last updated…