Question Number 307 by userid1 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}{te}^{−\mathrm{3}{t}} \mathrm{cos}\:{t}\:{dt} \\ $$ Commented by 123456 last updated on 20/Dec/14 $$\underset{\mathrm{0}} {\overset{\infty} {\int}}{te}^{−\mathrm{3}{t}}…
Question Number 65834 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:\:\:\forall\:\:{x},\:{y}\:\:>\mathrm{0}\:\:\:\:{B}\left({x},{y}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt}\:\:\:\:\:\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{show}\:{that}\:\:\forall\:{x}>\mathrm{0}\:\:\:\:\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right)\:\:\:\:{and}\:\:{lim}_{{n}−>\infty}…
Question Number 65837 by mathmax by abdo last updated on 04/Aug/19 $$\left.\mathrm{1}\right)\:{calculate}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}+{ix}}\:\:{and}\:\int_{−\infty} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}−{ix}} \\ $$$$\left.\mathrm{2}\right){deduce}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{ix}^{\mathrm{2}}…
Question Number 65828 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:{Let}\:{go}\:{toward}\:{a}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$$ \\ $$$${Part}\:\mathrm{1}\::\:\:{What}'{s}\:{that}\:{special}\:{factor}\:\: \\ $$$${Let}\:{n}\:,\:{p}\:{and}\:{k}\:{three}\:{integer}\:\:{different}\:{of}\:{zero} \\ $$$${We}\:\:{state}\:{J}_{{n},{k}} \left({p}\right)=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 131364 by EDWIN88 last updated on 04/Feb/21 $$ \\ $$$$\mathrm{How}\:\mathrm{can}\:\mathrm{I}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{volume}\: \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{region}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} +\mathrm{3}\:;\mathrm{x}=\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{x}=\mathrm{2}\:\mathrm{rotating}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}=\mathrm{7}\:\mathrm{using} \\ $$$$\mathrm{the}\:\mathrm{shell}\:\mathrm{method}. \\ $$ Answered by bramlexs22 last…
Question Number 65827 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $${Prove}\:{that}\:\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int_{\frac{\mathrm{1}}{\mathrm{6}}} ^{\frac{\mathrm{5}}{\mathrm{6}}} \:\:\frac{{dv}}{\left(\mathrm{1}−\:^{{v}} \sqrt{{u}}\:\right)^{{v}} }\right){du}={ln}\mathrm{2}−\mathrm{2}{ln}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right) \\ $$…
Question Number 65825 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $${let}\:{consider}\:\:{two}\:{real}\:{numbers}\:{p}\:{and}\:{such}\:{as}\:{p}^{\mathrm{2}} −{q}^{\mathrm{2}} ={pq} \\ $$$${Prove}\:{that} \\ $$$$\:\:\:{J}=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dv}}{\:^{{q}} \sqrt{\left(\mathrm{1}+\:^{{q}} \sqrt{{v}^{{p}}…
Question Number 65805 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:\:{Prove}\:{that}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+\left({tant}\right)^{{n}} }\:\:{does}\:{not}\:{depend}\:{of}\:{the}\:{term}\:{n} \\ $$$${deduces}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2035}}…
Question Number 261 by raj last updated on 25/Jan/15 $$\mathrm{If}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}{f}\left({t}\right){dt}={x}+\underset{{x}} {\overset{\mathrm{1}} {\int}}{tf}\left({t}\right){dt},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{f}\left(\mathrm{1}\right). \\ $$ Answered by prakash jain last updated on…
Question Number 260 by 9999 last updated on 25/Jan/15 $$\int_{−\mathrm{1}} ^{+\mathrm{1}} \mid\mathrm{1}−{x}\mid{dx}= \\ $$ Answered by 123456 last updated on 17/Dec/14 $$\mid\mathrm{1}−{x}\mid=\begin{cases}{\mathrm{1}−{x}}&{{x}\leqslant\mathrm{1}}\\{{x}−\mathrm{1}}&{{x}\geqslant\mathrm{1}}\end{cases} \\ $$$$\int_{−\mathrm{1}} ^{+\mathrm{1}}…