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Question Number 148026 Answers: 1 Comments: 0
$$ \\ $$$$\mathrm{A}\::=\int_{−\mathrm{1}} ^{\:\mathrm{0}} {e}^{\:{x}\:+\frac{\mathrm{1}}{{x}}} \:\:\&\:{a}\mathrm{A}+{e}^{{b}} =\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{{e}}\right)^{\:{x}+\frac{\mathrm{1}}{{x}}} \\ $$$$\:\:{than}\::\:\:{a}+\:{b}\:=?\:\:\:\:\:{a}\:,\:{b}\:\in\:\mathbb{Z} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Question Number 148020 Answers: 0 Comments: 0
$$\underset{\mathrm{k}=\mathrm{2}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \centerdot\frac{\mathrm{lnk}}{\mathrm{k}}=\gamma\mathrm{ln2}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}^{\mathrm{2}} \mathrm{2} \\ $$
Question Number 148000 Answers: 2 Comments: 0
$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$
Question Number 147884 Answers: 0 Comments: 0
$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{x}} {x}^{{a}} {dx}\:\:{a}>\mathrm{0} \\ $$$$ \\ $$
Question Number 147803 Answers: 0 Comments: 7
$$\int\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{4}−{x}^{\mathrm{4}} }}{dx} \\ $$
Question Number 147799 Answers: 1 Comments: 0
$$\int\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }}{dx} \\ $$
Question Number 147749 Answers: 1 Comments: 0
Question Number 147747 Answers: 0 Comments: 0
Question Number 147746 Answers: 1 Comments: 0
Question Number 147683 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{5}} \left(\mathrm{2x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:\mathrm{find}\:\int\:\mathrm{F}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\mathrm{en}\:\mathrm{deduire}\:\mathrm{la}\:\mathrm{decomposition}\:\mathrm{de}\:\mathrm{F}\:\mathrm{en}\:\mathrm{element}\:\mathrm{simples} \\ $$
Question Number 147682 Answers: 0 Comments: 0
$$\mathrm{decompose}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{n}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)}\:\mathrm{dans}\:\mathrm{C}\left(\mathrm{x}\right)\:\mathrm{puis}\:\mathrm{dans}\:\mathrm{R}\left(\mathrm{x}\right) \\ $$
Question Number 147680 Answers: 0 Comments: 2
$$\mathrm{find}\:\mathrm{by}\:\mathrm{residus}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dx} \\ $$
Question Number 147670 Answers: 1 Comments: 0
Question Number 147576 Answers: 1 Comments: 0
$$\left(\mathrm{1}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mid\mathrm{i}−\mathrm{j}\mid=? \\ $$$$\left(\mathrm{2}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{i}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{j}}=? \\ $$$$\left(\mathrm{3}\right)::\:\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}^{\mathrm{2}} } {\sum}}\left[\sqrt{\mathrm{i}}\right]=? \\ $$
Question Number 147487 Answers: 1 Comments: 0
$$ \\ $$$$ \\ $$$$\left({a}\:,\:\mathrm{2}{a}\:+\mathrm{1}\:\right]\cap\left[\:{a}^{\:\mathrm{2}} \:−{a}\:,\:{a}^{\:\mathrm{2}} +\:\mathrm{4}{a}\:+\mathrm{1}\:\right)\neq\:\varnothing \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:\in\:? \\ $$$$ \\ $$
Question Number 147477 Answers: 1 Comments: 1
Question Number 147399 Answers: 0 Comments: 0
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{e}^{\mathrm{2}{arctg}\left({u}\right)} }{\:\sqrt{{u}}} \\ $$
Question Number 147310 Answers: 1 Comments: 1
Question Number 147309 Answers: 2 Comments: 0
Question Number 147287 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{Advanced}\:\:\mathrm{Calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{C}{alculate}\:::\:\:\:\:\begin{cases}{\:\:\mathrm{i}\:::\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\mathrm{x}\right).\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\:\mathrm{dx}}\\{\:\:\mathrm{ii}\:::\:\:\:\:\:\mathrm{J}\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{Li}_{\:\mathrm{2}} \left(\:\mathrm{1}−\:\mathrm{x}^{\:\mathrm{2}} \right)\:=?}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Note}::\:\:\:\mathrm{Li}_{\mathrm{2}} \:\left(\mathrm{x}\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{\:\infty} {\sum}}\:\frac{\mathrm{x}^{\:{n}} }{{n}^{\:\mathrm{2}} }\:\:\:\:........\:\blacksquare\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\mathrm{m}.\mathrm{n}.... \\ $$$$ \\ $$
Question Number 147275 Answers: 1 Comments: 0
$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {e}^{{t}−\frac{{t}^{\mathrm{2}} }{\mathrm{2}}} {dt}\: \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({t}\right){dt}=\sqrt{{e}}−\mathrm{1} \\ $$
Question Number 147206 Answers: 0 Comments: 0
$$\mathrm{calculste}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$
Question Number 147203 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } }{\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}^{\mathrm{2}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{n}\geqslant\mathrm{1}\right) \\ $$$$\mathrm{nature}\:\mathrm{of}\:\Sigma\mathrm{U}_{\mathrm{n}} \:\mathrm{and}\:\Sigma\:\mathrm{nU}_{\mathrm{n}} \\ $$
Question Number 147202 Answers: 0 Comments: 0
$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}+\sqrt{\mathrm{2x}^{\mathrm{2}} +\mathrm{1}}}\mathrm{dx} \\ $$
Question Number 147166 Answers: 2 Comments: 0
$$\int_{\mathbb{R}} \mathrm{e}^{\mathrm{ixt}} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}} } \mathrm{dt}.. \\ $$
Question Number 147163 Answers: 0 Comments: 0
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