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Category: Integration

x-dx-

Question Number 133426 by metamorfose last updated on 22/Feb/21 $$\int\lfloor{x}\rfloor{dx}=?… \\ $$ Answered by MJS_new last updated on 22/Feb/21 $$\mathrm{for}\:{a}<{b}:\:\underset{{a}} {\overset{{b}} {\int}}\lfloor{x}\rfloor{dx}=\lfloor{a}\rfloor\underset{{a}} {\overset{\lceil{a}\rceil} {\int}}{dx}+\underset{\lceil{a}\rceil} {\overset{\lfloor{b}\rfloor−\mathrm{1}}…

Prove-that-m-Z-r-1-m-x-r-1-m-m-1-2-mx-2pi-m-1-2-mx-x-0-t-x-1-e-t-dt-x-gt-0-

Question Number 2340 by Yozzi last updated on 18/Nov/15 $${Prove}\:{that},\:\forall{m}\in\mathbb{Z}^{+} , \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{m}} {\prod}}\Gamma\left({x}+\frac{{r}−\mathrm{1}}{{m}}\right)={m}^{\frac{\mathrm{1}}{\mathrm{2}}−{mx}} \left(\mathrm{2}\pi\right)^{\frac{{m}−\mathrm{1}}{\mathrm{2}}} \Gamma\left({mx}\right). \\ $$$$\left\{\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} {e}^{−{t}} {dt},\:{x}>\mathrm{0}\right\} \\ $$…

find-dx-x-2-z-with-z-from-C-

Question Number 67851 by mathmax by abdo last updated on 01/Sep/19 $${find}\:\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} −{z}}\:\:{with}\:{z}\:{from}\:{C}\:. \\ $$ Commented by MJS last updated on 01/Sep/19 $$\mathrm{what}'\mathrm{s}\:\mathrm{the}\:\mathrm{problem}/\mathrm{mistake}\:\mathrm{if}\:\mathrm{we}\:\mathrm{just} \\ $$$$\mathrm{calculate}\:\mathrm{it}\:\mathrm{as}\:\mathrm{if}\:{z}\in\mathbb{R}?…

1-sec-x-dx-

Question Number 133360 by liberty last updated on 21/Feb/21 $$\int\:\sqrt{\mathrm{1}+\mathrm{sec}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$ Commented by som(math1967) last updated on 21/Feb/21 $$\int\sqrt{\frac{\mathrm{1}+{cosx}}{{cosx}}}{dx} \\ $$$$\sqrt{\mathrm{2}}\int\frac{{cos}\frac{{x}}{\mathrm{2}}}{\:\sqrt{\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}}{dx} \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}}}\int\frac{{cos}\frac{{x}}{\mathrm{2}}}{\:\sqrt{\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}}…

Define-a-curve-E-by-the-parametric-equations-x-t-g-1-t-h-1-t-f-1-u-du-y-t-g-2-t-h-2-t-f-2-u-du-where-h-1-g-1-h-2-and-g-2-are-different

Question Number 2284 by Yozzi last updated on 13/Nov/15 $${Define}\:{a}\:{curve}\:{E}\:{by}\:{the}\:{parametric} \\ $$$${equations} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\left({t}\right)=\int_{{g}_{\mathrm{1}} \left({t}\right)} ^{{h}_{\mathrm{1}} \left({t}\right)} {f}_{\mathrm{1}} \left({u}\right){du} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\left({t}\right)=\int_{{g}_{\mathrm{2}} \left({t}\right)} ^{{h}_{\mathrm{2}} \left({t}\right)} {f}_{\mathrm{2}}…