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Question Number 66061 Answers: 0 Comments: 0

let f(t) =∫_0 ^1 ((sinx)/(1+te^(−x^2 ) ))dx with ∣t∣<1 developp f at integr serie .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sinx}}{\mathrm{1}+{te}^{−{x}^{\mathrm{2}} } }{dx}\:\:\:\:{with}\:\mid{t}\mid<\mathrm{1} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 66060 Answers: 0 Comments: 3

let f(x) =∫_0 ^(π/4) (dt/(x+tant)) with x real 1) find aexplicit form of f(x) 2)find also g(x) =∫_0 ^(π/4) (dt/((x+tant)^2 )) 3)give f^((n)) (x)at form of integral 4)calculate ∫_0 ^(π/4) (dt/(2+tant)) and ∫_0 ^(π/4) (dt/((2+tant)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{{x}+{tant}}\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left({x}+{tant}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right){at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\mathrm{2}+{tant}}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left(\mathrm{2}+{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 66058 Answers: 2 Comments: 5

If x + (1/x)=1 find out value:− ((x^(20) +x^(17) +x^(14) +x^(11) )/(x^(17) +x^(14) +x^(11) +x^8 )) = ?

$${If}\:\:\:\:{x}\:+\:\frac{\mathrm{1}}{{x}}=\mathrm{1} \\ $$$$ \\ $$$${find}\:{out}\:{value}:− \\ $$$$ \\ $$$$\:\:\:\:\frac{{x}^{\mathrm{20}} +{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} }{{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} +{x}^{\mathrm{8}} }\:\:\:\:=\:\:\:\:? \\ $$

Question Number 66055 Answers: 1 Comments: 1

Question Number 66048 Answers: 0 Comments: 1

∫(x/(√(ln(1/x)))) dx

$$\int\frac{{x}}{\sqrt{{ln}\left(\mathrm{1}/{x}\right)}}\:{dx} \\ $$

Question Number 66046 Answers: 0 Comments: 2

find (dy/dx) if y=x^x^x help pls

$${find}\:\frac{{dy}}{{dx}}\:{if}\:{y}={x}^{{x}^{{x}} } \\ $$$${help}\:{pls} \\ $$

Question Number 66044 Answers: 1 Comments: 0

Question Number 66036 Answers: 1 Comments: 0

Question Number 66032 Answers: 0 Comments: 2

simplify w_n =(1+in)^n −(1−in)^n with n integr natural

$${simplify}\:\:{w}_{{n}} =\left(\mathrm{1}+{in}\right)^{{n}} −\left(\mathrm{1}−{in}\right)^{{n}} \:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 66019 Answers: 2 Comments: 1

lim_(x→∞) (1 + (2/x))^x =

$$\underset{{x}\rightarrow\infty} {{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{2}}{{x}}\right)^{{x}} \:= \\ $$

Question Number 66018 Answers: 1 Comments: 1

prove by mathematical induction that Σ_(r=1) ^n r(r + 1) = (n/3)(n + 1)(n + 2)

$${prove}\:{by}\:{mathematical}\:{induction}\:{that}\: \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right) \\ $$

Question Number 66017 Answers: 0 Comments: 0

show that the equation xe^x =1 has a root between 0.5 and 0.6 starting with 0.55 as a first approximate.

$${show}\:{that}\:{the}\:{equation}\:{xe}^{{x}} =\mathrm{1}\:{has}\:{a}\:{root}\:{between}\:\mathrm{0}.\mathrm{5}\:{and}\:\mathrm{0}.\mathrm{6}\:{starting} \\ $$$${with}\:\mathrm{0}.\mathrm{55}\:{as}\:{a}\:{first}\:{approximate}. \\ $$

Question Number 66016 Answers: 1 Comments: 6

Evaluate a. ∫_1 ^2 (lnx)^2 dx b. ∫_0 ^(π/6) sin^2 x cos^3 xdx

$${Evaluate}\:\:\: \\ $$$${a}.\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({lnx}\right)^{\mathrm{2}} {dx} \\ $$$${b}.\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{3}} {xdx} \\ $$

Question Number 66010 Answers: 0 Comments: 1

Using Σ_(n=1) ^∞ (1/(n!))=e , prove that lim_(n→∞) (1+(1/n))^n =e

$${Using}\:\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{{n}!}={e}\:,\:{prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} ={e} \\ $$

Question Number 66005 Answers: 0 Comments: 0

solve x^2 y^(′′) +xy^′ −3y =4e^(−x)

$${solve}\:\:\:{x}^{\mathrm{2}} {y}^{''} \:+{xy}^{'} −\mathrm{3}{y}\:=\mathrm{4}{e}^{−{x}} \\ $$

Question Number 66004 Answers: 1 Comments: 0

x^x^x^⋰ =3 find the value of x

$${x}^{{x}^{{x}^{\iddots} } } =\mathrm{3} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x} \\ $$

Question Number 65992 Answers: 1 Comments: 1

x = (1/(3∙1!)) + (1/(4∙2!)) + (1/(5∙3!)) + … + (1/(1002∙1000!)) x∙1000! = ?

$${x}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{1}!}\:+\:\frac{\mathrm{1}}{\mathrm{4}\centerdot\mathrm{2}!}\:+\:\frac{\mathrm{1}}{\mathrm{5}\centerdot\mathrm{3}!}\:+\:\ldots\:+\:\frac{\mathrm{1}}{\mathrm{1002}\centerdot\mathrm{1000}!} \\ $$$${x}\centerdot\mathrm{1000}!\:\:=\:\:? \\ $$

Question Number 65988 Answers: 3 Comments: 1

∫dx/x^2 −x+1

$$\int{dx}/{x}^{\mathrm{2}} −{x}+\mathrm{1} \\ $$

Question Number 66041 Answers: 0 Comments: 0

Question Number 65983 Answers: 1 Comments: 1

Simplify (1+ 2i(√2))^7 − (1 +2i)^7

$$\:{Simplify}\: \\ $$$$\:\:\:\left(\mathrm{1}+\:\mathrm{2}{i}\sqrt{\mathrm{2}}\right)^{\mathrm{7}} \:−\:\left(\mathrm{1}\:+\mathrm{2}{i}\right)^{\mathrm{7}} \\ $$

Question Number 65972 Answers: 1 Comments: 0

If ((log_2 a)/(log_3 b))=m and ((log_3 a)/(log_2 b))=n a>1 and b>1 then (m/n)=... a.log_2 3 b. log_3 2 c. log_4 9 d. (log_2 3)^2 e. (log_3 2)^2

Question Number 65971 Answers: 2 Comments: 0

If (a^2 /b^2 )=12 then log(^3 (√(b/a)))=.. a. −2 b. −1 c. 0 d. 1 e. 2

$$\mathrm{If}\:\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{12}\:\mathrm{then}\:\mathrm{log}\left(\:^{\mathrm{3}} \sqrt{\frac{{b}}{{a}}}\right)=.. \\ $$$${a}.\:−\mathrm{2} \\ $$$${b}.\:−\mathrm{1} \\ $$$${c}.\:\mathrm{0} \\ $$$${d}.\:\mathrm{1} \\ $$$${e}.\:\mathrm{2} \\ $$

Question Number 65970 Answers: 1 Comments: 0

log_5 (√(27))×log_9 125+log_(16) 12=... a. ((61)/(36)) b. (9/4) c. ((61)/(20)) d. ((41)/(12)) e. (7/2)

$$\mathrm{log}_{\mathrm{5}} \sqrt{\mathrm{27}}×\mathrm{log}_{\mathrm{9}} \mathrm{125}+\mathrm{log}_{\mathrm{16}} \mathrm{12}=... \\ $$$${a}.\:\frac{\mathrm{61}}{\mathrm{36}} \\ $$$${b}.\:\frac{\mathrm{9}}{\mathrm{4}} \\ $$$${c}.\:\frac{\mathrm{61}}{\mathrm{20}} \\ $$$${d}.\:\frac{\mathrm{41}}{\mathrm{12}} \\ $$$${e}.\:\frac{\mathrm{7}}{\mathrm{2}} \\ $$

Question Number 65961 Answers: 1 Comments: 0

Let (d/dx)(F(x)) = (e^(sin x) /x) , x>0. If ∫_( 1) ^4 ((2 e^(sin x^2 ) )/x) dx = F(k)−F(1), then one of the possible values of k is

$$\mathrm{Let}\:\frac{{d}}{{dx}}\left({F}\left({x}\right)\right)\:=\:\frac{{e}^{\mathrm{sin}\:{x}} }{{x}}\:,\:{x}>\mathrm{0}. \\ $$$$\mathrm{If}\:\underset{\:\mathrm{1}} {\overset{\mathrm{4}} {\int}}\:\:\frac{\mathrm{2}\:{e}^{\mathrm{sin}\:{x}^{\mathrm{2}} } }{{x}}\:{dx}\:=\:{F}\left({k}\right)−{F}\left(\mathrm{1}\right),\:\mathrm{then}\:\mathrm{one} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\:{k}\:\:\mathrm{is} \\ $$

Question Number 65951 Answers: 0 Comments: 1

Question Number 65950 Answers: 4 Comments: 0

∫_0 ^( π/2) tan^3 xdx = ?

$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{tan}\:^{\mathrm{3}} {xdx}\:=\:? \\ $$

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