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Question Number 57329 Answers: 1 Comments: 1

∫_(−1) ^2 ∣x∣ ⌊x⌋ dx = ?

$$\underset{−\mathrm{1}} {\int}\overset{\mathrm{2}} {\:}\:\mid{x}\mid\:\lfloor{x}\rfloor\:{dx}\:\:=\:\:\:? \\ $$

Question Number 57328 Answers: 1 Comments: 0

Question Number 57325 Answers: 0 Comments: 1

calculate ∫_0 ^(π/2) ((ln(1+sinx))/(sinx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{ln}\left(\mathrm{1}+{sinx}\right)}{{sinx}}{dx} \\ $$

Question Number 57324 Answers: 0 Comments: 0

we want to find the vslue of I =∫_0 ^1 ((ln(1+x))/(1+x^2 )) dx let A=∫∫_W (x/((1+x^2 )(1+xy)))dxdy with W=[0,1]^2 calculate A by two method and conclude the value of I .

$${we}\:{want}\:{to}\:{find}\:{the}\:{vslue}\:{of} \\ $$$${I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{let} \\ $$$${A}=\int\int_{{W}} \frac{{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{xy}\right)}{dxdy} \\ $$$${with}\:{W}=\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} \\ $$$${calculate}\:{A}\:{by}\:{two}\:{method}\:{and} \\ $$$${conclude}\:{the}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 57323 Answers: 0 Comments: 1

calculate ∫∫_D ((x+y)/(3+(√(x^2 +y^2 ))))dxdy with D={(x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0 ,y≥0}

$${calculate}\:\int\int_{{D}} \:\:\frac{{x}+{y}}{\mathrm{3}+\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{2}\right. \\ $$$$\left.{and}\:{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\right\} \\ $$

Question Number 57321 Answers: 1 Comments: 1

calculate ∫∫_D (x−y)(√(x^2 +y^2 ))dxdy with D ={ (x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0}

$${calculate}\:\int\int_{{D}} \left({x}−{y}\right)\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{D}\:=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{2}\:{and}\:{x}\geqslant\mathrm{0}\right\} \\ $$

Question Number 57320 Answers: 1 Comments: 1

calculate ∫∫_D xy e^(−x^2 −y^2 ) dxdy with D={(x,y)∈R^2 / 0≤x≤2 and 1≤y≤3}

$${calculate}\:\int\int_{{D}} {xy}\:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\:{and}\right. \\ $$$$\left.\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\right\} \\ $$

Question Number 57319 Answers: 1 Comments: 1

calculate ∫∫_D e^(x−y) dxdy with D={(x,y)∈R^2 /∣x∣<1 and 0≤y≤1}

$${calculate}\:\int\int_{{D}} \:{e}^{{x}−{y}} \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\mid{x}\mid<\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$

Question Number 57310 Answers: 0 Comments: 1

Question Number 57309 Answers: 0 Comments: 0

Question Number 57306 Answers: 1 Comments: 2

Question Number 57302 Answers: 1 Comments: 0

Question Number 57296 Answers: 1 Comments: 1

Question Number 57289 Answers: 0 Comments: 3

Tangents are drawn to x^2 +y^2 =16 from the point P(0,h).These tangents meet the x−axis at A and B. If area of ΔPAB is minimum then find value of h ?

$${Tangents}\:{are}\:{drawn}\:{to}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{16}\:{from} \\ $$$${the}\:{point}\:{P}\left(\mathrm{0},{h}\right).{These}\:{tangents}\:{meet} \\ $$$${the}\:{x}−{axis}\:{at}\:{A}\:{and}\:{B}.\:{If}\:{area}\:{of}\:\Delta{PAB} \\ $$$${is}\:{minimum}\:{then}\:{find}\:{value}\:{of}\:{h}\:? \\ $$

Question Number 57284 Answers: 1 Comments: 1

y is varies directly as the square of x and inversely as z. if x is inceased by 10% and z is decreased by 20%, find the percentage change in y.

$${y}\:{is}\:{varies}\:{directly}\:{as}\:{the}\:{square}\:{of}\:{x}\:{and} \\ $$$${inversely}\:{as}\:{z}. \\ $$$${if}\:{x}\:{is}\:{inceased}\:{by}\:\mathrm{10\%}\:{and}\:{z}\:\:{is}\: \\ $$$${decreased}\:{by}\:\mathrm{20\%},\:{find}\:{the}\:{percentage} \\ $$$${change}\:{in}\:{y}. \\ $$

Question Number 57283 Answers: 1 Comments: 1

Question Number 57269 Answers: 2 Comments: 6

If A>0,B>0, and A+B=(π/3) , then maximum value of tanAtanB is ?

$${If}\:{A}>\mathrm{0},{B}>\mathrm{0},\:{and}\:{A}+{B}=\frac{\pi}{\mathrm{3}}\:,\:{then} \\ $$$${maximum}\:{value}\:{of}\:{tanAtanB}\:{is}\:? \\ $$

Question Number 57253 Answers: 0 Comments: 4

lim_(x→0) ((e^x + e^(−x) )/x)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\:\:\:\:\frac{\mathrm{e}^{\mathrm{x}} \:+\:\mathrm{e}^{−\mathrm{x}} }{\mathrm{x}}\:\:\:\:\:\: \\ $$

Question Number 57251 Answers: 1 Comments: 0

Question Number 57245 Answers: 1 Comments: 0

((a^8 +a^4 +1)/(a^4 +a^2 +1))=?

$$\frac{\boldsymbol{\mathrm{a}}^{\mathrm{8}} +\boldsymbol{\mathrm{a}}^{\mathrm{4}} +\mathrm{1}}{\boldsymbol{\mathrm{a}}^{\mathrm{4}} +\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\mathrm{1}}=? \\ $$

Question Number 57244 Answers: 0 Comments: 1

Question Number 57228 Answers: 0 Comments: 1

find f(x) =∫_1 ^2 ((ln(1+xt))/t^2 ) dt with x>0

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{2}} \:\frac{{ln}\left(\mathrm{1}+{xt}\right)}{{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 57227 Answers: 0 Comments: 0

let f(α)=∫_0 ^1 ((arctan(αx))/(1+αx^2 )) dx with α real 1) find f(α) interms of α 2) find the values of ∫_0 ^1 ((arctan(2x))/(1+2x^2 )) dx and ∫_0 ^1 ((arctan(4x))/(1+4x^2 ))dx

$${let}\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+\alpha{x}^{\mathrm{2}} }\:{dx}\:\:\:{with}\:\alpha\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left(\alpha\right)\:{interms}\:{of}\:\alpha \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }\:{dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{4}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 57226 Answers: 0 Comments: 0

calculate A_n =∫_0 ^1 x^n (√((1−x)/(1+x)))dx with n integr natural

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx}\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 57224 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((3t^2 −5t +1)/((t+1)(t+2)(2t+3)))dt

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{5}{t}\:+\mathrm{1}}{\left({t}+\mathrm{1}\right)\left({t}+\mathrm{2}\right)\left(\mathrm{2}{t}+\mathrm{3}\right)}{dt} \\ $$

Question Number 57225 Answers: 0 Comments: 2

1)calculate f(a) =∫_0 ^a ((2x−1)/((x^2 −x+3)(x^2 +1)))dx 1) calculate f(1)and f(2)

$$\left.\mathrm{1}\right){calculate}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{{a}} \:\:\:\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}^{\mathrm{2}} \:−{x}+\mathrm{3}\right)\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)}{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\mathrm{1}\right){and}\:{f}\left(\mathrm{2}\right) \\ $$

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