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Category: Integration

2-0-1-tan-1-x-dx-

Question Number 197744 by mathlove last updated on 27/Sep/23 $$\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} {tan}^{−\mathrm{1}} {x}\:{dx}=? \\ $$ Answered by witcher3 last updated on 27/Sep/23 $$\int\mathrm{2tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\mathrm{dx}=\mathrm{2xtan}^{−\mathrm{1}} \left(\mathrm{x}\right)−\int\frac{\mathrm{2x}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}}…

0-pi-2-dx-3-tan-x-

Question Number 197436 by horsebrand11 last updated on 17/Sep/23 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{dx}}{\mathrm{3}+\mathrm{tan}\:\mathrm{x}}\:=? \\ $$ Answered by Frix last updated on 17/Sep/23 $$\int\frac{{dx}}{\mathrm{3}+\mathrm{tan}\:{x}}\:\overset{{t}=\mathrm{tan}\:{x}} {=}\:\int\frac{{dt}}{\left({t}+\mathrm{3}\right)\left({t}^{\mathrm{2}} +\mathrm{1}\right)}= \\…

Does-anyone-know-how-to-prove-this-V-dxdydz-1-x-4-y-4-z-4-4-1-4-4-4-where-V-is-the-unit-cube-0-1-3-Thankyou-

Question Number 197376 by megrex last updated on 15/Sep/23 $${Does}\:{anyone}\:{know}\:{how}\:{to}\:{prove}\:{this}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\int\int\int_{{V}} \:\frac{{dxdydz}}{\mathrm{1}+{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} }\:=\frac{\Gamma^{\mathrm{4}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{4}^{\mathrm{4}} } \\ $$$${where}\:{V}\:{is}\:{the}\:{unit}\:{cube}\:\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} \\ $$$${Thankyou}. \\ $$$$ \\…