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Category: Integration

evaluate-0-tanh-1-2-z-csch-z-z-dz-1-Complex-integral-2-Feynman-trick-

Question Number 213081 by issac last updated on 30/Oct/24 $$\mathrm{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\:\:\frac{\mathrm{tanh}\left(\frac{\mathrm{1}}{\mathrm{2}}{z}\right)\mathrm{csch}\left({z}\right)}{{z}}\mathrm{d}{z} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Complex}\:\mathrm{integral} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Feynman}\:\mathrm{trick} \\ $$ Answered by Berbere last updated…

3x-2-5x-2-2x-3-dx-

Question Number 213098 by MathematicalUser2357 last updated on 30/Oct/24 $$\int\frac{\mathrm{3}{x}+\mathrm{2}}{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}}\mathrm{d}{x}=? \\ $$ Answered by issac last updated on 30/Oct/24 $$\mathrm{Hmmmmm}….. \\ $$$$\frac{\mathrm{3}{x}+\mathrm{2}}{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}}=\frac{\mathrm{3}\left(\mathrm{10}{x}+\mathrm{2}\right)}{\mathrm{10}\left(\mathrm{5}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\right)}+\frac{\mathrm{7}}{\mathrm{5}\left(\mathrm{5}{x}^{\mathrm{2}}…

Question-213074

Question Number 213074 by Spillover last updated on 29/Oct/24 Answered by MrGaster last updated on 29/Oct/24 $$=\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \left[−\frac{\mathrm{cos}\left({x}+{y}^{\mathrm{2}} +{z}^{\mathrm{3}} \right)}{\mathrm{3}{z}^{\mathrm{2}} }\right]_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{3}}}…

Question-212899

Question Number 212899 by vasil92 last updated on 26/Oct/24 Answered by MrGaster last updated on 02/Nov/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{{n}} }{dx}=\mathrm{1} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[{x}−\frac{{x}^{{n}−\mathrm{1}} }{\left({n}+\mathrm{1}\right)\centerdot^{\mathrm{2}}…

m-1-3-1-2x-2-7-dx-k-find-the-value-of-the-constant-m-and-k-

Question Number 212635 by Nadirhashim last updated on 19/Oct/24 $$\:\:\boldsymbol{{m}}\leqslant\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}\:}{\:\sqrt{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{7}\:}}\:.\boldsymbol{{dx}}\leqslant\boldsymbol{{k}}\:\boldsymbol{{find}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{constant}} \\ $$$$\:\:\boldsymbol{{m}}\:\boldsymbol{{and}}\:\boldsymbol{{k}} \\ $$ Commented by Ghisom last updated on…

let-f-x-1-x-a-x-b-x-c-let-a-b-c-R-a-lt-b-lt-c-D-f-x-a-b-c-prove-a-b-f-x-dx-c-f-x-dx-

Question Number 212626 by Ghisom last updated on 19/Oct/24 $$\mathrm{let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\left({x}−{a}\right)\left({x}−{b}\right)\left({x}−{c}\right)}} \\ $$$$\mathrm{let}\:{a},\:{b},\:{c}\:\in\mathbb{R}\:\wedge{a}<{b}<{c} \\ $$$$\Rightarrow\:{D}\left({f}\left({x}\right)\right)=\left({a},\:{b}\right)\cup\left({c},\:\infty\right) \\ $$$$\mathrm{prove}\:\underset{{a}} {\overset{{b}} {\int}}{f}\left({x}\right){dx}=\underset{{c}} {\overset{\infty} {\int}}{f}\left({x}\right){dx} \\ $$ Answered by MrGaster…

lim-n-2-1-n-1-2n-1-1-n-1-1-2n-e-y-2-dy-2n-1-2n-e-y-2-dy-

Question Number 212553 by MrGaster last updated on 17/Oct/24 $$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt[{{n}}]{\mathrm{2}}−\mathrm{1}}{\:\sqrt[{{n}}]{\mathrm{2}{n}+\mathrm{1}}}\mid\int_{\mathrm{1}} ^{\frac{\mathrm{1}}{\mathrm{2}{n}}} {e}^{−{y}^{\mathrm{2}} } {dy}+\ldots+\int^{\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}{n}}} {e}^{−{y}^{\mathrm{2}} } {dy}\mid=? \\ $$$$ \\ $$ Terms…