Question and Answers Forum
All Questions Topic List
IntegrationQuestion and Answers: Page 300
Question Number 32338 Answers: 0 Comments: 0
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt}. \\ $$
Question Number 32337 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right){calculate}\:\int_{{a}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:\:. \\ $$
Question Number 32323 Answers: 1 Comments: 0
$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} {f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Find}\:{f}\left({x}\right) \\ $$
Question Number 32304 Answers: 0 Comments: 0
$${find}\:{lim}_{{x}\rightarrow+\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{{t}^{\mathrm{2}} } {dt}\:\:. \\ $$
Question Number 32302 Answers: 1 Comments: 0
$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\:\frac{{dx}}{{x}\:+{x}\sqrt{{x}}}\:. \\ $$
Question Number 32301 Answers: 0 Comments: 1
$${calculate}\:\int_{\mathrm{1}} ^{{e}} \:{ln}\left(\mathrm{1}+\sqrt{{x}}\right){dx}\:. \\ $$
Question Number 32269 Answers: 1 Comments: 0
$${find}\:\int\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$
Question Number 32258 Answers: 2 Comments: 0
$${find} \\ $$$$\int\:\frac{\mathrm{1}}{\mathrm{2}−{x}^{\mathrm{2}} }\:{dx} \\ $$
Question Number 32206 Answers: 0 Comments: 0
$$\mathrm{Find}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\underset{\mathrm{k}−\mathrm{1}} {\overset{\mathrm{k}} {\int}}\mathrm{x}^{−\mathrm{x}} \:\mathrm{dx}\right)\:. \\ $$$$ \\ $$
Question Number 32139 Answers: 0 Comments: 4
$${F}\boldsymbol{{ind}}\:\boldsymbol{{the}} \\ $$$$\int\:\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$
Question Number 32045 Answers: 0 Comments: 0
$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}} \:{sin}^{{n}} {t}\:{dt}\:\:. \\ $$
Question Number 32044 Answers: 0 Comments: 1
$${fimd}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:{dt}\:. \\ $$
Question Number 32043 Answers: 0 Comments: 0
$${let}\:{f}\left({x}\right)=\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{dt}}{{lnt}}\:\:{with}\:{x}>\mathrm{0}\:{and}\:{x}\neq\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{1}\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{xdt}}{{tlnt}}\:\leqslant{f}\left({x}\right)\leqslant\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{x}^{\mathrm{2}} {dt}}{{tlnt}}\:\:{after} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:. \\ $$
Question Number 32040 Answers: 0 Comments: 2
$${let}\:{give}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{x}\:{tant}}\:\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{tant}}{\left(\mathrm{1}+{xtant}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){give}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tant}}{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{tant}\right)^{\mathrm{2}} }\:{dt}\:. \\ $$
Question Number 32039 Answers: 0 Comments: 3
$${a}>−\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{tan}^{\mathrm{2}} {t}}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+{atan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+\mathrm{2}{tan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }{dt}.\: \\ $$
Question Number 32034 Answers: 0 Comments: 0
$${let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+...+{x}^{{n}} }\:\:{study}\:{the}\:{convergence}\:{of} \\ $$$$\Sigma\:{u}_{{n}} \:\:. \\ $$
Question Number 32031 Answers: 0 Comments: 1
$${let}\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} \left({xlnx}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} \left({xlnx}\right){dx}\:\:. \\ $$
Question Number 32029 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right)\:{dx}\:\:{with}\:\:\alpha>\mathrm{0}\:. \\ $$
Question Number 32026 Answers: 0 Comments: 1
$${let}\:\alpha>\mathrm{0}\:{prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\alpha}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\alpha−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:. \\ $$
Question Number 31974 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right){find}\:{I}\left({p},{q}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{t}^{{p}} \:\left(\mathrm{1}−{t}\right)^{{q}} \:{dt}\:\:{with}\:{pand}\:{q}\:{integrs} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:\:{I}_{\left({n},{n}\right)} \\ $$
Question Number 31970 Answers: 0 Comments: 0
$${find}\:{the}\:{nature}\:{of}\:\:\int_{\mathrm{2}} ^{\infty} \:\:\frac{{e}^{−{x}} }{\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:{dx}\:. \\ $$
Question Number 31969 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{4}} }\right){arctant}\:{dt}. \\ $$
Question Number 31968 Answers: 0 Comments: 0
$${find}\:\:\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} {x}\sqrt{\left({x}−\mathrm{2}\right)\left(\sqrt{\mathrm{5}}−{x}\right)}\:{dx}\:. \\ $$
Question Number 31967 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}{dx}\:. \\ $$
Question Number 31951 Answers: 1 Comments: 0
$${Evaluate}\:\int\mathrm{sin}\:\sqrt{{x}}{dx} \\ $$
Question Number 31858 Answers: 0 Comments: 1
$$\int\frac{{sinx}}{{x}}{dx} \\ $$
Pg 295 Pg 296 Pg 297 Pg 298 Pg 299 Pg 300 Pg 301 Pg 302 Pg 303 Pg 304
Terms of Service
Privacy Policy
Contact: info@tinkutara.com