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let f(α) =∫_0 ^1 x(√(x^2 −x+α))dx with α>(1/4) 1) explicit f(α) 2) calculate g(α) =∫_0 ^1 ((xdx)/(√(x^2 −x+α))) 3) find the value of intehrals ∫_0 ^1 x(√(x^2 −x+(√2)))dx snd ∫_0 ^1 ((xdx)/(√(x^2 −x+(√2))))

$${let}\:{f}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {x}\sqrt{{x}^{\mathrm{2}} −{x}+\alpha}{dx}\:\:\:\:\:\:{with}\:\alpha>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{explicit}\:\:{f}\left(\alpha\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{xdx}}{\sqrt{{x}^{\mathrm{2}} −{x}+\alpha}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{intehrals}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}\sqrt{{x}^{\mathrm{2}} −{x}+\sqrt{\mathrm{2}}}{dx}\:{snd} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{xdx}}{\sqrt{{x}^{\mathrm{2}} −{x}+\sqrt{\mathrm{2}}}} \\ $$

Question Number 92081 Answers: 0 Comments: 0

find ∫_0 ^1 (x^3 −3)(√(x^2 +2x+5))dx

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}^{\mathrm{3}} −\mathrm{3}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$

Question Number 92080 Answers: 0 Comments: 0

calculate lim_(x→0) ((sin(2shx) −sh(2sinx))/(e^x −1))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{sin}\left(\mathrm{2}{shx}\right)\:−{sh}\left(\mathrm{2}{sinx}\right)}{{e}^{{x}} −\mathrm{1}} \\ $$

Question Number 92079 Answers: 0 Comments: 1

find lim_(x→0) ((e^(sin^2 x) −e^(x^3 −2x) )/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{e}^{{sin}^{\mathrm{2}} {x}} −{e}^{{x}^{\mathrm{3}} −\mathrm{2}{x}} }{{x}^{\mathrm{2}} } \\ $$

Question Number 92078 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) ((cos(arctan(2x+1)))/(x^2 +x+1))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{cos}\left({arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$

Question Number 92056 Answers: 0 Comments: 2

lim_(x→0^+ ) ((ln(x))/(cot x))

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{ln}\left({x}\right)}{\mathrm{cot}\:{x}} \\ $$

Question Number 92055 Answers: 0 Comments: 3

∫(((x+1)/((x^2 +4x+5)^2 )))dx

$$\int\left(\frac{\mathrm{x}+\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{5}\right)^{\mathrm{2}} }\right)\mathrm{dx} \\ $$

Question Number 92054 Answers: 0 Comments: 0

tan^2 ((π/7))+tan^2 (((2π)/7))+tan^2 (((3π)/7)) ?

$$\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\pi}{\mathrm{7}}\right)+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:? \\ $$

Question Number 92041 Answers: 1 Comments: 0

solve :[ 2yy′′=1+(y′)^2 ] if y(0)=2 , y′(0)=−1

$${solve}\::\left[\:\:\mathrm{2}{yy}''=\mathrm{1}+\left({y}'\right)^{\mathrm{2}} \:\right]\:{if}\:{y}\left(\mathrm{0}\right)=\mathrm{2}\:,\:{y}'\left(\mathrm{0}\right)=−\mathrm{1} \\ $$$$ \\ $$

Question Number 92040 Answers: 1 Comments: 0

y^(′′) +(y′)^2 +y=0 y(0)=−(1/2) , y′(0)=−1 help me sir

$${y}^{''} +\left({y}'\right)^{\mathrm{2}} +{y}=\mathrm{0}\:\:\:{y}\left(\mathrm{0}\right)=−\frac{\mathrm{1}}{\mathrm{2}}\:,\:{y}'\left(\mathrm{0}\right)=−\mathrm{1} \\ $$$${help}\:{me}\:{sir}\: \\ $$

Question Number 92036 Answers: 0 Comments: 2

2^3^2 =? A. 64 B. 512

$$\mathrm{2}^{\mathrm{3}^{\mathrm{2}} } =? \\ $$$${A}.\:\mathrm{64} \\ $$$${B}.\:\mathrm{512} \\ $$

Question Number 92032 Answers: 0 Comments: 0

lim_(x→∞) ((2^(4n−1) ∗(n!)^2 )/((2n+1)∗[2n p(n)]^2 ))

$$\underset{{x}\rightarrow\infty} {{lim}}\frac{\mathrm{2}^{\mathrm{4}{n}−\mathrm{1}} \ast\left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\ast\left[\mathrm{2}{n}\:{p}\left({n}\right)\right]^{\mathrm{2}} } \\ $$

Question Number 92016 Answers: 1 Comments: 0

Question Number 92013 Answers: 1 Comments: 10

Solve: 2^x + 3^y = 72 ..... (i) 2^y + 3^x = 108 ..... (ii)

$$\mathrm{Solve}: \\ $$$$\:\:\:\mathrm{2}^{\mathrm{x}} \:\:+\:\:\mathrm{3}^{\mathrm{y}} \:\:\:=\:\:\mathrm{72}\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\mathrm{2}^{\mathrm{y}} \:\:+\:\:\mathrm{3}^{\mathrm{x}} \:\:\:=\:\:\mathrm{108}\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$

Question Number 92010 Answers: 1 Comments: 0

if log_6 30 = a and log_(24) 15 = b log_(12) 60 = ?

$$\mathrm{if}\:\mathrm{log}_{\mathrm{6}} \mathrm{30}\:=\:{a}\:\mathrm{and}\:\mathrm{log}_{\mathrm{24}} \mathrm{15}\:=\:{b} \\ $$$$\mathrm{log}_{\mathrm{12}} \mathrm{60}\:=\:? \\ $$

Question Number 92008 Answers: 0 Comments: 1

Sum of infinite series: 1 + (3/4) + ((3.5)/(4.8)) + ((3.5.7)/(4.8.12)) + ... is ?

$$\mathrm{Sum}\:\mathrm{of}\:\mathrm{infinite}\:\mathrm{series}:\:\:\mathrm{1}\:\:+\:\:\frac{\mathrm{3}}{\mathrm{4}}\:\:+\:\:\frac{\mathrm{3}.\mathrm{5}}{\mathrm{4}.\mathrm{8}}\:\:+\:\:\frac{\mathrm{3}.\mathrm{5}.\mathrm{7}}{\mathrm{4}.\mathrm{8}.\mathrm{12}}\:\:+\:\:...\:\:\:\:\mathrm{is}\:? \\ $$

Question Number 92003 Answers: 0 Comments: 2

The sum to n terms of the series (3/1^2 ) + (5/(1^2 +2^2 )) + (7/(1^2 +2^2 +3^2 )) + .... is

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{to}\:{n}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\frac{\mathrm{3}}{\mathrm{1}^{\mathrm{2}} }\:+\:\frac{\mathrm{5}}{\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} }\:+\:\frac{\mathrm{7}}{\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }\:+\:....\:\mathrm{is} \\ $$

Question Number 91996 Answers: 0 Comments: 2

∫(arcsinx)^2 dx=?

$$\int\left(\mathrm{arcsinx}\right)^{\mathrm{2}} \mathrm{dx}=? \\ $$

Question Number 91995 Answers: 0 Comments: 1

hello what is the metric of schwarzchild dynamics.

$${hello}\:{what}\:{is}\:{the}\:{metric}\:{of}\:{schwarzchild}\:{dynamics}. \\ $$

Question Number 92025 Answers: 2 Comments: 1

∫(((x−1)/(x^2 −x−1)))dx

$$\int\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{1}}\right)\mathrm{dx} \\ $$

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