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Question Number 116299 Answers: 0 Comments: 0
$$\:\:\:...\:\:{advanced}\:\:{math}\:... \\ $$$$\:\:\:\:\:\:\:{evaluate}\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left[\frac{\mathrm{1}}{{ln}\left({x}\right)}\:+\frac{\mathrm{1}}{\mathrm{1}−{x}}\:\right]^{\mathrm{2}} {dx}=??? \\ $$$$\:\:\:\:\:\:\:{m}.{n} \\ $$
Question Number 116290 Answers: 0 Comments: 1
Question Number 116301 Answers: 2 Comments: 0
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{for}\:\mathrm{a} \\ $$$$\mathrm{given}\:\mathrm{line}\:\mathrm{to}\:\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{a}\:\mathrm{curve} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{not}\:\mathrm{to}\:\mathrm{intersect}\:\mathrm{a}\:\mathrm{curve}\: \\ $$
Question Number 116281 Answers: 2 Comments: 0
$$\mathrm{Prove}\:\mathrm{that}\:\:\:\mathrm{sin}\:\mathrm{10}°\:\mathrm{sin}\:\mathrm{30}°\:\mathrm{sin}\:\mathrm{50}°\:\mathrm{sin}\:\mathrm{70}°. \\ $$
Question Number 116284 Answers: 3 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}\:{I}\:... \\ $$$$\:\:\:\:\:{evaluate}\::\: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} {log}\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{tan}\left({x}\right)\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$
Question Number 116279 Answers: 2 Comments: 0
$$\int\:\frac{{xe}^{{x}} }{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$
Question Number 116276 Answers: 0 Comments: 0
Question Number 116272 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({x}^{\mathrm{2}} +{ln}^{\mathrm{2}} \left({cos}\left({x}\right)\right)\right){dx}=\pi{ln}\left({ln}\left(\mathrm{2}\right)\right) \\ $$$${posted}\:{Quation}\: \\ $$$${not}\:{solved}\:{yet}\:{i}\:{hop}\:{someon}\:{Giv}\:{idea}\:{for} \\ $$$${this}\:{one}\:{thank}\:{you} \\ $$
Question Number 116271 Answers: 2 Comments: 0
$$\int_{−\mathrm{1}} ^{\:\mathrm{0}} \left[{ln}\left({ln}\left(\mathrm{1}+{x}\right)\right)\right]^{\mathrm{2}} {dx} \\ $$
Question Number 116267 Answers: 2 Comments: 0
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{parabola}\:\mathrm{that} \\ $$$$\mathrm{passes}\:\mathrm{through}\:\mathrm{points}\:\left(\mathrm{1},\mathrm{2}\right)\:,\left(\mathrm{3},\mathrm{4}\right)\: \\ $$$$\mathrm{and}\:\mathrm{tangents}\:\mathrm{to}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=−\mathrm{x}+\frac{\mathrm{25}}{\mathrm{3}} \\ $$
Question Number 116259 Answers: 1 Comments: 0
Question Number 116253 Answers: 0 Comments: 0
$$\left(\mathrm{1}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right)\:\:\:=\:\:\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right).\:\mathrm{x}_{\mathrm{i}} ^{\mathrm{k}} \:\:\:=\:\:\:\mathrm{x}^{\mathrm{k}} ,\:\:\:\:\:\:\:\:\mathrm{k}\:\leqslant\:\mathrm{n} \\ $$
Question Number 116252 Answers: 2 Comments: 0
$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} }{\mathrm{e}^{\mathrm{x}} −\mathrm{e}^{−\mathrm{x}} }\:=\:? \\ $$
Question Number 116250 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\mathrm{explicite}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{n}\left[\mathrm{x}\right]} \mathrm{cos}\left(\mathrm{3}\left[\mathrm{x}\right]\right)\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$$$\left.\mathrm{3}\right)\mathrm{find}\:\mathrm{nsture}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \\ $$
Question Number 116249 Answers: 0 Comments: 0
$$\mathrm{find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}\mathrm{dx} \\ $$
Question Number 116248 Answers: 0 Comments: 0
$$\mathrm{calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{2}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} −\mathrm{x}\:+\mathrm{1}}\mathrm{dx} \\ $$
Question Number 116247 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{ch}\left(\mathrm{cos}\left(\mathrm{2x}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{9}}\mathrm{dx}\:\mathrm{and} \\ $$$$\mathrm{J}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{ch}\left(\mathrm{2x}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{9}}\mathrm{dx} \\ $$
Question Number 116245 Answers: 3 Comments: 0
$$\mathrm{calculate}\:\:\int_{\mathrm{1}} ^{\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{2x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}} } \\ $$
Question Number 116239 Answers: 4 Comments: 0
$$\mathrm{a}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{x}−\mathrm{axis}\:,\:\mathrm{y}−\mathrm{axis} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}−\mathrm{4y}+\mathrm{6}=\mathrm{0}. \\ $$$$\mathrm{what}\:\mathrm{its}\:\mathrm{the}\:\mathrm{equation}? \\ $$
Question Number 116238 Answers: 1 Comments: 0
$$\mathrm{How}\:\mathrm{long}\:\mathrm{does}\:\mathrm{it}\:\mathrm{take}\:\mathrm{for}\:\mathrm{the}\:\mathrm{disintegration}\:\mathrm{of} \\ $$$$\frac{\mathrm{3}}{\mathrm{10}}\:\mathrm{of}\:\mathrm{an}\:\mathrm{atom}\:\mathrm{with}\:\mathrm{radioactive}\:\mathrm{constant}\:\lambda\:? \\ $$
Question Number 116237 Answers: 2 Comments: 0
$$\int\:\sqrt{\mathrm{5cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{4}}\:\mathrm{dx}\:? \\ $$
Question Number 116231 Answers: 1 Comments: 0
$$\int\:\mathrm{sec}\:\mathrm{x}\:\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3}}\:\mathrm{dx}\:? \\ $$
Question Number 116221 Answers: 3 Comments: 0
$$\mathrm{Given}\:\alpha,\beta\:\mathrm{and}\:\varphi\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{pq}\:=\:\mathrm{0}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\varphi}+\frac{\varphi}{\beta}+\frac{\alpha}{\varphi}+\frac{\varphi}{\alpha}=? \\ $$
Question Number 116216 Answers: 1 Comments: 0
$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$
Question Number 116192 Answers: 2 Comments: 1
Question Number 116184 Answers: 1 Comments: 0
$$\mathrm{How}\:\mathrm{many}\:\mathrm{6}-\mathrm{digits}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{1}\:\mathrm{to}\:\mathrm{9}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:\mathrm{each} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the}\:\mathrm{number}\:\mathrm{appears}\:\mathrm{at}\:\mathrm{least} \\ $$$$\mathrm{twice}?\:\left[\mathrm{For}\:\mathrm{instance}:\:\mathrm{121233},\mathrm{122221},\mathrm{777777}\:\mathrm{and}\:\mathrm{etc}.\right] \\ $$
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