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Question Number 115014 Answers: 1 Comments: 0

if I_n =∫_x ^(π/2) xcos^n xdx,where n≻1 show that I_n =((n(n−1)I_(n−2) −1)/n^2 ) and then evaluate ∫_x ^(π/2) xcos^8 xdx

$${if}\:{I}_{{n}} =\int_{{x}} ^{\frac{\pi}{\mathrm{2}}} {x}\mathrm{cos}^{{n}} {xdx},{where}\:{n}\succ\mathrm{1}\:{show} \\ $$$${that}\:{I}_{{n}} =\frac{{n}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} −\mathrm{1}}{{n}^{\mathrm{2}} }\:{and}\:{then} \\ $$$${evaluate}\:\:\int_{{x}} ^{\frac{\pi}{\mathrm{2}}} {x}\mathrm{cos}^{\mathrm{8}} {xdx} \\ $$

Question Number 115013 Answers: 0 Comments: 0

consider the change in the direction of a curve W=f(θ) between point A and B. Derive from first principle an expression for the radius of curvature R for the hyperbola

$${consider}\:{the}\:{change}\:{in}\:{the}\:{direction}\:{of} \\ $$$${a}\:{curve}\:{W}={f}\left(\theta\right)\:{between}\:{point}\:{A}\:{and} \\ $$$${B}.\:{Derive}\:{from}\:{first}\:{principle}\:{an} \\ $$$${expression}\:{for}\:{the}\:{radius}\:{of}\:{curvature} \\ $$$${R}\:{for}\:{the}\:{hyperbola} \\ $$

Question Number 115011 Answers: 0 Comments: 0

given the quadratic function f(x)=x+2+px+q where p and q are?integer.s let a,b, and c distinc integers such that 2^(2020) evenly divides f(a),f(b),and f(c),but 2^(1000) does not divide b−a and also does not divide c−a.show that 2^(1021) just divide b−c?

$${given}\:{the}\:{quadratic}\:{function}\: \\ $$$${f}\left({x}\right)={x}+\mathrm{2}+{px}+{q}\:{where}\:{p}\:{and}\:{q}\:{are}?{integer}.{s} \\ $$$${let}\:{a},{b},\:{and}\:{c}\:{distinc}\:{integers}\:{such}\:{that} \\ $$$$\mathrm{2}^{\mathrm{2020}} \:\:{evenly}\:{divides}\:{f}\left({a}\right),{f}\left({b}\right),{and}\:{f}\left({c}\right),{but} \\ $$$$\mathrm{2}^{\mathrm{1000}} \:{does}\:{not}\:{divide}\:{b}−{a}\:{and}\:{also}\:{does}\:{not} \\ $$$${divide}\:{c}−{a}.{show}\:{that}\:\mathrm{2}^{\mathrm{1021}} \:{just}\:{divide}\:{b}−{c}? \\ $$

Question Number 115009 Answers: 2 Comments: 0

∫_C (e^z /(1−cos z))dz ; C:∣z∣=1

$$\int_{\mathrm{C}} \frac{{e}^{{z}} }{\mathrm{1}−\mathrm{cos}\:{z}}{dz}\:;\:\mathrm{C}:\mid{z}\mid=\mathrm{1} \\ $$

Question Number 115000 Answers: 2 Comments: 0

....nice math... if y =(cos(2x))^(−(1/2)) then prove :: y+y^(′′) = 3y^5 ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:{math}... \\ $$$$ \\ $$$$\:{if}\:\:{y}\:=\left({cos}\left(\mathrm{2}{x}\right)\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \:{then} \\ $$$${prove}\:::\:\:\:{y}+{y}^{''} =\:\mathrm{3}{y}^{\mathrm{5}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$

Question Number 115095 Answers: 2 Comments: 0

Question Number 114990 Answers: 0 Comments: 0

Question Number 114988 Answers: 2 Comments: 0

Prove tan^2 x=sin^2 x+sec^2 x

$$\mathrm{Prove}\:\mathrm{tan}^{\mathrm{2}} {x}=\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{sec}^{\mathrm{2}} {x} \\ $$

Question Number 114981 Answers: 2 Comments: 2

Without L′Hopital (1)lim_(x→1) ((x(x+(1/x))^5 −32)/(x−1)) =? (2) lim_(x→∞) (√(2x+(√(2x+(√(2x+(√(2x+(√(...)))))))))) −(√(2x)) = ?

$${Without}\:{L}'{Hopital} \\ $$$$\:\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{{x}\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{5}} −\mathrm{32}}{{x}−\mathrm{1}}\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{...}}}}}\:−\sqrt{\mathrm{2}{x}}\:=\:? \\ $$

Question Number 114980 Answers: 1 Comments: 0

Question Number 114978 Answers: 2 Comments: 0

Solve x^2 dy + y(x+y)dx=0

$$ \\ $$$$\:\:{Solve}\: \\ $$$$\:{x}^{\mathrm{2}} {dy}\:+\:{y}\left({x}+{y}\right){dx}=\mathrm{0} \\ $$

Question Number 114975 Answers: 1 Comments: 0

.... nice mathematics.... show that :: Φ = ∫_0 ^( 1) li_2 (x)dx = (π^2 /6) −1 ✓ m.n.july.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:{nice}\:\:{mathematics}....\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{show}\:\:{that}\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {li}_{\mathrm{2}} \left({x}\right){dx}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 114967 Answers: 0 Comments: 1

Question Number 114965 Answers: 1 Comments: 0

find all irational numbers x such that x 2+20x+20 and x^(3 ) −2020x+1 both is a rasional number.

$${find}\:{all}\:{irational}\:{numbers}\:{x}\:{such}\:{that}\:{x} \\ $$$$\mathrm{2}+\mathrm{20}{x}+\mathrm{20}\:{and}\:{x}^{\mathrm{3}\:} −\mathrm{2020}{x}+\mathrm{1}\:\:\:{both}\:{is}\:{a} \\ $$$${rasional}\:{number}. \\ $$

Question Number 114996 Answers: 2 Comments: 1

...nice mathematics... prove that::: i:: Σ_(n=1) ^∞ (1/(sinh^2 (πn))) =(1/6) −(1/(2π)) ✓ ii:: Σ_(n=1) ^∞ (n/(e^(2πn) −1))=(1/(24)) −(1/(8π)) ✓✓ iii::Σ_(n=1) ^∞ (1/( nsinh(πn))) =(π/(12))−((ln(2))/4) ✓✓✓ .... M..n..july..1970 ....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{mathematics}...\: \\ $$$$\:\:\:\:{prove}\:{that}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{sinh}^{\mathrm{2}} \left(\pi{n}\right)}\:=\frac{\mathrm{1}}{\mathrm{6}}\:−\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{8}\pi}\:\:\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{iii}::\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\:{nsinh}\left(\pi{n}\right)}\:=\frac{\pi}{\mathrm{12}}−\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{4}}\:\checkmark\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:\:\:\mathscr{M}..{n}..{july}..\mathrm{1970}\:.... \\ $$$$ \\ $$

Question Number 114959 Answers: 2 Comments: 0

long time question proposed by math abdo ∫_0 ^∞ ((lnx)/(1+x^2 +x^4 ))dx

$${long}\:{time}\:{question}\:{proposed}\:{by} \\ $$$${math}\:{abdo} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}{x}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$

Question Number 114956 Answers: 0 Comments: 5

Question Number 114950 Answers: 0 Comments: 0

note the tringle ABC is not isosceles with the elevtions of AA1,BB1, and CC1.suppose BA amd CA respectively point at BB1 and CC1 so that A1BA is pependiculer to BB1 and A1CA perpendiculer to CC1.the read and BC lines intersect at the TA point. define in the same way TB and TC poits. prove that TA,TB,and TC are collinear.

$${note}\:{the}\:{tringle}\:{ABC}\:\:{is}\:{not}\:{isosceles}\:{with}\: \\ $$$${the}\:{elevtions}\:{of}\:{AA}\mathrm{1},{BB}\mathrm{1},\:{and}\:{CC}\mathrm{1}.{suppose} \\ $$$${BA}\:\:{amd}\:{CA}\:{respectively}\:{point}\:{at}\:{BB}\mathrm{1}\:{and} \\ $$$${CC}\mathrm{1}\:{so}\:{that}\:{A}\mathrm{1}{BA}\:{is}\:{pependiculer}\:{to}\:{BB}\mathrm{1} \\ $$$${and}\:{A}\mathrm{1}{CA}\:{perpendiculer}\:{to}\:{CC}\mathrm{1}.{the}\:{read}\: \\ $$$${and}\:{BC}\:{lines}\:{intersect}\:{at}\:{the}\:{TA}\:{point}. \\ $$$${define}\:{in}\:{the}\:{same}\:{way}\:\:{TB}\:{and}\:{TC}\:{poits}. \\ $$$${prove}\:{that}\:{TA},{TB},{and}\:{TC}\:{are}\:{collinear}. \\ $$

Question Number 114945 Answers: 1 Comments: 0