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Question Number 77464 Answers: 3 Comments: 7

∫_( 0) ^(π/2) ln (2 cos x) dx = ?

$$\underset{\:\:\mathrm{0}} {\int}\:\overset{\frac{\pi}{\mathrm{2}}} {\:}\mathrm{ln}\:\left(\mathrm{2}\:\mathrm{cos}\:{x}\right)\:{dx}\:\:=\:\:? \\ $$

Question Number 77462 Answers: 0 Comments: 0

Question Number 77459 Answers: 1 Comments: 0

the greatest value of n so that only one value of k satisfies (8/(15))<(n/(n+k))<(7/(13)) is

$$ \\ $$$$ \\ $$$${the}\:{greatest}\:{value}\:{of}\:{n}\:{so}\:{that}\: \\ $$$${only}\:{one}\:{value}\:{of}\:{k}\:{satisfies}\: \\ $$$$\frac{\mathrm{8}}{\mathrm{15}}<\frac{{n}}{{n}+{k}}<\frac{\mathrm{7}}{\mathrm{13}}\:{is}\: \\ $$

Question Number 77452 Answers: 0 Comments: 0

∫e^(x^3 +x^2 −1) (3x^4 +2x^2 +2x)dx

$$\int\mathrm{e}^{\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \left(\mathrm{3x}^{\mathrm{4}} +\mathrm{2x}^{\mathrm{2}} +\mathrm{2x}\right)\mathrm{dx} \\ $$

Question Number 77451 Answers: 1 Comments: 1

Question Number 77447 Answers: 1 Comments: 0

∫ tan(x) tan(2x) tan(3x) dx

$$\int\:\mathrm{tan}\left(\mathrm{x}\right)\:\mathrm{tan}\left(\mathrm{2x}\right)\:\mathrm{tan}\left(\mathrm{3x}\right)\:\mathrm{dx} \\ $$

Question Number 77442 Answers: 1 Comments: 0

prove that lim_(x→∞) (∣((x^x^2 (x+2)^((x+1)^2 ) )/((x+1)^(2x^2 +2x+1) ))∣)=e

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

Question Number 77425 Answers: 1 Comments: 2

Question Number 77424 Answers: 0 Comments: 0

∫_0 ^∞ e^((e^x −1)^t (A)) dx A and t are constant

$$\int_{\mathrm{0}} ^{\infty} {e}^{\left({e}^{{x}} −\mathrm{1}\right)^{{t}} \:\left({A}\right)} \:{dx} \\ $$$${A}\:{and}\:{t}\:{are}\:{constant} \\ $$

Question Number 77422 Answers: 1 Comments: 1

can solve ∫ (dx/(x^(17) −1)) via elementary calculus?

$$\mathrm{can}\:\mathrm{solve}\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{17}} −\mathrm{1}}\:\mathrm{via}\: \\ $$$$\mathrm{elementary}\:\mathrm{calculus}? \\ $$

Question Number 77420 Answers: 0 Comments: 10

Question Number 77412 Answers: 0 Comments: 7

Question Number 77394 Answers: 1 Comments: 4

a 6 digit number formed with the first number is 2 and contains 4 equal number. many number can be made are

$$\mathrm{a}\:\mathrm{6}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{formed}\: \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{first}\:\mathrm{number}\:\mathrm{is}\:\mathrm{2}\:\mathrm{and}\: \\ $$$$\mathrm{contains}\:\mathrm{4}\:\mathrm{equal}\:\mathrm{number}.\:\mathrm{many} \\ $$$$\mathrm{number}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{are}\: \\ $$

Question Number 77390 Answers: 2 Comments: 1

Question Number 77379 Answers: 0 Comments: 0

Question Number 77377 Answers: 0 Comments: 0

Question Number 77367 Answers: 1 Comments: 1

let the cercle (x+1)^(2 ) +(y−3)^2 =9 and the point A(4,1) vrrify that A is out of circle and determine the equation of two tangentes to circle wich passes by point A.

$${let}\:{the}\:{cercle}\:\:\left({x}+\mathrm{1}\right)^{\mathrm{2}\:} +\left({y}−\mathrm{3}\right)^{\mathrm{2}} =\mathrm{9} \\ $$$${and}\:{the}\:{point}\:\:{A}\left(\mathrm{4},\mathrm{1}\right) \\ $$$${vrrify}\:{that}\:\:{A}\:\:{is}\:{out}\:{of}\:{circle} \\ $$$${and}\:\:{determine}\:{the}\:{equation}\:{of} \\ $$$${two}\:{tangentes}\:{to}\:{circle}\:{wich} \\ $$$${passes}\:{by}\:{point}\:{A}. \\ $$

Question Number 77356 Answers: 1 Comments: 0

The plan is provided with an orthonormal reference ( O.I.J). the following points are given A(1,2) B(−2,3) C(1,9). We assume that the point O is the barycenter of the point A,B,C. →O=bar{(A;3),(B;1),(C;−1)} Question 1 knowing that 3MA^2 +MB^2 −MC^2 =3MO^2 +3OA^2 +OB^2 −OC^2 Determine and construct the set of points M on the plane such as 3MA^2 +MB^2 −MC^2 =−42

$$\mathrm{The}\:\mathrm{plan}\:\mathrm{is}\:\mathrm{provided}\:\mathrm{with}\:\mathrm{an}\: \\ $$$$\mathrm{orthonormal}\:\mathrm{reference}\:\left(\:\mathrm{O}.\mathrm{I}.\mathrm{J}\right). \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{points}\:\mathrm{are}\:\mathrm{given} \\ $$$$\mathrm{A}\left(\mathrm{1},\mathrm{2}\right)\:\mathrm{B}\left(−\mathrm{2},\mathrm{3}\right)\:\mathrm{C}\left(\mathrm{1},\mathrm{9}\right). \\ $$$$\mathrm{We}\:\mathrm{assume}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{barycenter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{A},\mathrm{B},\mathrm{C}. \\ $$$$\rightarrow\mathrm{O}=\mathrm{bar}\left\{\left(\mathrm{A};\mathrm{3}\right),\left(\mathrm{B};\mathrm{1}\right),\left(\mathrm{C};−\mathrm{1}\right)\right\} \\ $$$$ \\ $$$$\mathrm{Question}\:\mathrm{1} \\ $$$$\mathrm{knowing}\:\mathrm{that} \\ $$$$\mathrm{3MA}^{\mathrm{2}} +\mathrm{MB}^{\mathrm{2}} −\mathrm{MC}^{\mathrm{2}} =\mathrm{3MO}^{\mathrm{2}} +\mathrm{3OA}^{\mathrm{2}} +\mathrm{OB}^{\mathrm{2}} −\mathrm{OC}^{\mathrm{2}} \\ $$$$\mathrm{Determine}\:\mathrm{and}\:\mathrm{construct}\:\mathrm{the}\:\mathrm{set}\: \\ $$$$\mathrm{of}\:\mathrm{points}\:\mathrm{M}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{such}\:\mathrm{as} \\ $$$$\mathrm{3MA}^{\mathrm{2}} +\mathrm{MB}^{\mathrm{2}} −\mathrm{MC}^{\mathrm{2}} =−\mathrm{42} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 77347 Answers: 2 Comments: 1

if ∫sin(f(x))dx=g(x) ∫cos(f(x))dx=?

$$\mathrm{if}\:\int\mathrm{sin}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\int\mathrm{cos}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=? \\ $$

Question Number 77346 Answers: 0 Comments: 3