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AllQuestion and Answers: Page 124

Question Number 210031 Answers: 0 Comments: 0

∫_0 ^(π/2) ((1−xcot(x))/x^2 )dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}−\boldsymbol{\mathrm{xcot}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}} \\ $$

Question Number 210025 Answers: 0 Comments: 1

Question Number 210017 Answers: 0 Comments: 0

MATH−WHIZZKID using kamke find the genral solution for the differential equation 1. x^2 y′′+x^2 y′−2y=0 −−−−−−−−− solve this using forbenius mtd 1.x^2 y′′+(x^3 −3x)y′+(4−2x)y=0 −−−−−−−− solve the differential eqn by power series 1. y′′−2xy′+2py=0 −−−−−−−−− use perseval′s theorem to ∫_0 ^∞ ((cos^2 (𝛂(𝛑/2)))/((1−𝛂^2 )^2 ))dx. −−−−−−−−−− evaluate this integral by contour integration 1. ∫_0 ^∞ ((cos^2 (𝛂(𝛑/2)))/((1−𝛂^2 )^2 ))dx. −−−−−−−−− ∮_c ((1+e^(i𝛑z) )/((z−1)^2 (z+1)^2 ))dz c−upper half plane klipto−quanta⊎

$$ \\ $$$$\boldsymbol{\mathrm{MATH}}−\boldsymbol{\mathrm{WHIZZKID}} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{kamke}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{genral}} \\ $$$$\boldsymbol{\mathrm{solution}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}''+\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}'−\mathrm{2}\boldsymbol{\mathrm{y}}=\mathrm{0} \\ $$$$−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{forbenius}}\:\boldsymbol{\mathrm{mtd}} \\ $$$$\mathrm{1}.\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}''+\left(\boldsymbol{\mathrm{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{y}}'+\left(\mathrm{4}−\mathrm{2}\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{y}}=\mathrm{0} \\ $$$$−−−−−−−− \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{eqn}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{power}}\:\boldsymbol{\mathrm{series}} \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{y}}''−\mathrm{2}\boldsymbol{\mathrm{xy}}'+\mathrm{2}\boldsymbol{\mathrm{py}}=\mathrm{0} \\ $$$$−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{use}}\:\boldsymbol{\mathrm{perseval}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{theorem}}\:\boldsymbol{\mathrm{to}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\boldsymbol{\alpha}\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)}{\left(\mathrm{1}−\boldsymbol{\alpha}^{\mathrm{2}} \right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}. \\ $$$$−−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{evaluate}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{integral}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{contour}}\:\boldsymbol{\mathrm{integration}} \\ $$$$\mathrm{1}.\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\boldsymbol{\alpha}\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)}{\left(\mathrm{1}−\boldsymbol{\alpha}^{\mathrm{2}} \right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}. \\ $$$$−−−−−−−−− \\ $$$$\oint_{\boldsymbol{\mathrm{c}}} \frac{\mathrm{1}+\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{i}\pi\mathrm{z}}} }{\left(\boldsymbol{\mathrm{z}}−\mathrm{1}\right)^{\mathrm{2}} \left(\boldsymbol{\mathrm{z}}+\mathrm{1}\right)^{\mathrm{2}} }\boldsymbol{\mathrm{dz}} \\ $$$$\boldsymbol{\mathrm{c}}−\mathrm{upper}\:\mathrm{half}\:\mathrm{plane} \\ $$$$\boldsymbol{\mathrm{klipto}}−\boldsymbol{\mathrm{quanta}}\biguplus \\ $$

Question Number 210014 Answers: 2 Comments: 3

Question Number 210013 Answers: 1 Comments: 0

If f(x)= x^2 +ax+b. if f(1)= 3 and and one of the roots of the eqiation f(x)= 0 doubles the other, find the positive values of a and b.

$$\:{If}\:{f}\left({x}\right)=\:{x}^{\mathrm{2}} +{ax}+{b}.\:{if}\:{f}\left(\mathrm{1}\right)=\:\mathrm{3}\:{and}\: \\ $$$$\:{and}\:{one}\:{of}\:{the}\:{roots}\:{of}\:{the}\:{eqiation} \\ $$$$\:{f}\left({x}\right)=\:\mathrm{0}\:{doubles}\:{the}\:{other},\:{find}\:{the} \\ $$$$\:{positive}\:{values}\:{of}\:{a}\:{and}\:{b}. \\ $$

Question Number 210011 Answers: 0 Comments: 0

Find: ∫_0 ^( 1) ((ln (cos (((πx)/2))))/(x^2 + x)) dx = ?

$$\mathrm{Find}:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\:\frac{\mathrm{ln}\:\left(\mathrm{cos}\:\left(\frac{\pi\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}}\:\mathrm{dx}\:\:=\:\:? \\ $$

Question Number 209999 Answers: 3 Comments: 0

lim_(x→0) ((e^x −1)/( (√(1−cosx)))) =?

$$\:\:\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{e}}^{\boldsymbol{{x}}} −\mathrm{1}}{\:\sqrt{\mathrm{1}−\boldsymbol{{cosx}}}}\:=? \\ $$

Question Number 209991 Answers: 2 Comments: 0

((10^(log _3 (6)) . 15^(log _3 ((2/3))) )/(6^(log _3 ((2/3))) . 5^(log _3 ((4/3))) )) =?

$$\:\:\:\:\:\frac{\mathrm{10}^{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{6}\right)} .\:\mathrm{15}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} }{\mathrm{6}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)} .\:\mathrm{5}^{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{4}}{\mathrm{3}}\right)} }\:=?\: \\ $$

Question Number 209988 Answers: 0 Comments: 0

Question Number 209986 Answers: 1 Comments: 0

Solve ax^3 −bx(√x) +c=0 (a, b, c)∈R^3 and x∈R (the value of x for a=1, b=9,c=8)

$$\mathrm{Solve}\: \\ $$$$\:\boldsymbol{\mathrm{ax}}^{\mathrm{3}} −\boldsymbol{\mathrm{bx}}\sqrt{\boldsymbol{\mathrm{x}}}\:+\boldsymbol{\mathrm{c}}=\mathrm{0}\:\:\:\:\: \\ $$$$\:\left(\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}},\:\boldsymbol{\mathrm{c}}\right)\in\mathbb{R}^{\mathrm{3}} \:\:\:\:\mathrm{and}\:\boldsymbol{\mathrm{x}}\in\mathbb{R} \\ $$$$\left(\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{x}}\:\boldsymbol{{for}}\:\boldsymbol{{a}}=\mathrm{1},\:\:\boldsymbol{{b}}=\mathrm{9},\boldsymbol{{c}}=\mathrm{8}\right) \\ $$

Question Number 209980 Answers: 0 Comments: 7

determiner h ? CD=20 AB=30 h1=25

$$\mathrm{determiner}\:\mathrm{h}\:? \\ $$$$\boldsymbol{\mathrm{CD}}=\mathrm{20}\:\:\:\:\boldsymbol{\mathrm{AB}}=\mathrm{30} \\ $$$$\boldsymbol{\mathrm{h}}\mathrm{1}=\mathrm{25} \\ $$$$ \\ $$

Question Number 209976 Answers: 0 Comments: 1

Question Number 209975 Answers: 1 Comments: 0

Question Number 209974 Answers: 0 Comments: 0

Question Number 209972 Answers: 1 Comments: 0

Question Number 209965 Answers: 1 Comments: 0

Question Number 209960 Answers: 2 Comments: 1

Determiner Aire (ABH) AH⊥CE

$$\mathrm{Determiner}\:\:\mathrm{Aire}\:\left(\boldsymbol{\mathrm{ABH}}\right) \\ $$$$\:\:\:\mathrm{AH}\bot\mathrm{CE} \\ $$

Question Number 209956 Answers: 3 Comments: 0

Find the maximum value of 7cosA + 24sinA + 32

Find the maximum value of 7cosA + 24sinA + 32

Question Number 209944 Answers: 0 Comments: 1

Question Number 209937 Answers: 1 Comments: 0

Question Number 209932 Answers: 4 Comments: 0

Question Number 209929 Answers: 0 Comments: 0

Question Number 209926 Answers: 1 Comments: 3

lim_(n→∞) (1/(3n+1))+(1/(3n+2))+...+(1/(4n))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{2}}+...+\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$

Question Number 209924 Answers: 0 Comments: 0

If f(x)=(x!)∙(x!!)∙(x!!!) find (d/dx)(f(x))=?

$$\boldsymbol{{If}}\:\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\left(\boldsymbol{{x}}!\right)\centerdot\left(\boldsymbol{{x}}!!\right)\centerdot\left(\boldsymbol{{x}}!!!\right)\:\: \\ $$$$\boldsymbol{{find}}\:\:\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)=? \\ $$

Question Number 209923 Answers: 1 Comments: 0

Solve: ∫((sin(x!))/(x!))dx

$$\boldsymbol{{Solve}}:\:\int\frac{\boldsymbol{{sin}}\left(\boldsymbol{{x}}!\right)}{\boldsymbol{{x}}!}\boldsymbol{{dx}} \\ $$

Question Number 209918 Answers: 1 Comments: 0

Find: ∫_0 ^( ∞) (({x}^([x]) )/([x] + 1)) dx = ? {x} → fractional part [x] → full part

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\left\{\mathrm{x}\right\}^{\left[\boldsymbol{\mathrm{x}}\right]} }{\left[\mathrm{x}\right]\:+\:\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$$$\left\{\mathrm{x}\right\}\:\rightarrow\:\mathrm{fractional}\:\mathrm{part} \\ $$$$\left[\mathrm{x}\right]\:\:\:\rightarrow\:\mathrm{full}\:\mathrm{part} \\ $$

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