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Category: Integration

prove-that-f-x-dx-1-such-that-f-x-1-n-n-2-1-2-1-x-2-n-1-2-1-n-and-n-2-1-2-0-x-n-2-1-1-x-3-2-dx-

Question Number 55364 by peter frank last updated on 22/Feb/19 $$\mathrm{prove}\:\mathrm{that}\:\int_{−\infty\:} ^{\infty} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{1} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)}\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{n}\right)} \\ $$$$\mathrm{and}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\frac{\mathrm{n}}{\mathrm{2}}−\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\mathrm{dx} \\ $$ Terms…

1-2-sin-3x-x-2-2-dx-1-2-3-1-sin-4t-t-2-3-4-dt-

Question Number 55360 by rahul 19 last updated on 22/Feb/19 $$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \sqrt{\mathrm{sin}\:\left(\mathrm{3}{x}−{x}^{\mathrm{2}} −\mathrm{2}\right)}{dx}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{3}} ^{\mathrm{1}} \sqrt{{sin}\left(\frac{\mathrm{4}{t}−{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{4}}\right)}{dt}\:\:=? \\ $$ Commented by rahul 19 last updated…

Nice-calculus-evaluate-S-n-1-2-2n-1-cos-pi-2-n-M-N-1970-

Question Number 120875 by mnjuly1970 last updated on 03/Nov/20 $$\:\:\:\:\:\:\:\:\:\:…\:\mathscr{N}{ice}\:{calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:{S}=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}^{−\mathrm{2}{n}} }{\mathrm{1}+{cos}\left(\frac{\pi}{\mathrm{2}^{{n}} }\right)}\right)=??\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\mathscr{M}.\mathscr{N}.\mathrm{1970}… \\ $$ Terms…

3x-3x-3-7-dx-

Question Number 55310 by afachri last updated on 21/Feb/19 $$ \\ $$$$\int\:\mathrm{3}{x}\sqrt{\:\mathrm{3}{x}^{\mathrm{3}} +\:\mathrm{7}}\:\:{dx}\:=\:\:.\:.\:.\:. \\ $$ Commented by MJS last updated on 22/Feb/19 $$\mathrm{where}\:\mathrm{does}\:\mathrm{this}\:\mathrm{come}\:\mathrm{from}? \\ $$$$\mathrm{I}\:\mathrm{tried}\:\mathrm{everything}\:\mathrm{I}\:\mathrm{know}\:\mathrm{but}\:\mathrm{it}\:\mathrm{seems}…

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

Question Number 186352 by normans last updated on 03/Feb/23 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\:\frac{\boldsymbol{{tan}}^{−\mathrm{1}} \:\left(\boldsymbol{{x}}\right)\:+\:\mathrm{2}}{\boldsymbol{{x}}^{\mathrm{2}} }\:\:\boldsymbol{{dx}}\:\:\:\:\:\:\: \\ $$$$ \\ $$ Answered by MJS_new last updated…

calculatef-a-1-a-x-2-arctan-a-x-dx-2-calculate-1-1-2-x-2-arctan-2-x-dx-

Question Number 55282 by Abdo msup. last updated on 20/Feb/19 $${calculatef}\left({a}\right)=\:\:\int\:\:\:\left(\mathrm{1}+\frac{{a}}{{x}^{\mathrm{2}} }\right){arctan}\left(\frac{{a}}{{x}}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{+\infty} \left(\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }\right){arctan}\left(\frac{\mathrm{2}}{{x}}\right){dx}\:. \\ $$ Commented by maxmathsup by imad last…