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Category: Integration

x-5-x-3-1-dx-

Question Number 11982 by tawa last updated on 08/Apr/17 $$\int\mathrm{x}^{\mathrm{5}} \left(\sqrt{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}}\right)\:\mathrm{dx} \\ $$ Answered by ajfour last updated on 08/Apr/17 $${I}=\frac{\mathrm{1}}{\mathrm{3}}\int{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:\left(\mathrm{3}{x}^{\mathrm{2}} {dx}\right)…

Lobachevsky-Integral-0-sin-2-tan-x-x-2-dx-pi-2-

Question Number 143051 by mnjuly1970 last updated on 09/Jun/21 $$\:\:\:\:\:\:\:_{\ast\ast\ast\ast\ast} ::\:\:{Lobachevsky}\:{Integral}\:::_{\ast\ast\ast\ast\ast} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{s}{in}^{\mathrm{2}} \left(\:{tan}\left({x}\right)\right)}{{x}^{\:\mathrm{2}} }{dx}\overset{?} {=}\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:………. \\ $$ Answered by Olaf_Thorendsen…

dx-5-4x-x-2-is-this-answer-correct-ln-1-4-x-5-ln6-1-x-C-

Question Number 11937 by tawa last updated on 05/Apr/17 $$\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{5}\:+\:\mathrm{4x}\:−\:\mathrm{x}^{\mathrm{2}} }}\: \\ $$$$ \\ $$$$ \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{answer}\:\mathrm{correct}\:?\:\:\:\:\:\:\:\:\:\:\:−\mathrm{ln}\left[\mathrm{1}/\mathrm{4}\left(\mathrm{x}\:−\:\mathrm{5}\right)\:−\:\mathrm{ln6}\left(−\:\mathrm{1}\:−\:\mathrm{x}\right)\right]\:+\:\mathrm{C} \\ $$ Commented by ridwan balatif last updated…

tan-x-tan-2x-tan-3x-dx-

Question Number 77447 by TawaTawa last updated on 06/Jan/20 $$\int\:\mathrm{tan}\left(\mathrm{x}\right)\:\mathrm{tan}\left(\mathrm{2x}\right)\:\mathrm{tan}\left(\mathrm{3x}\right)\:\mathrm{dx} \\ $$ Answered by peter frank last updated on 06/Jan/20 $$\mathrm{tan}\:\mathrm{3}{x}=\frac{\mathrm{tan}\:{x}+\mathrm{tan}\:\mathrm{2}{x}}{\mathrm{1}−\mathrm{tan}\:{x}\mathrm{tan}\:\mathrm{2}{x}} \\ $$$$\mathrm{tan}\:\mathrm{3}{x}−\mathrm{tan}{x}\:\mathrm{tan2}{x}\:\mathrm{tan}\:\mathrm{3}{x}=\mathrm{tan}{x}+\:\mathrm{tan}\:\mathrm{2}{x} \\ $$$$\mathrm{tan}{x}\:\mathrm{tan2}{x}\:\mathrm{tan}\:\mathrm{3}{x}=\mathrm{tan3}{x}−\:\mathrm{tan}\:\mathrm{2}{x}−\mathrm{tan}{x}\:…