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Question Number 78021    Answers: 0   Comments: 7

lim_(x→0) (1/x^2 )[∫^(x^2 +(π/3)) _(π/3) ((cos x)/x) dx ] =

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\left[\underset{\frac{\pi}{\mathrm{3}}} {\int}^{{x}^{\mathrm{2}} +\frac{\pi}{\mathrm{3}}} \frac{\mathrm{cos}\:{x}}{{x}}\:{dx}\:\right]\:= \\ $$

Question Number 78013    Answers: 1   Comments: 0

if : 30x^4 −((15)/8)= ∫_t ^x g(u)du find g(t).

$${if}\::\:\mathrm{30}{x}^{\mathrm{4}} −\frac{\mathrm{15}}{\mathrm{8}}=\:\underset{{t}} {\overset{{x}} {\int}}\:{g}\left({u}\right){du} \\ $$$${find}\:{g}\left({t}\right). \\ $$

Question Number 77995    Answers: 0   Comments: 3

calculate ∫_(−∞) ^(+∞) ((arctan(2x+1))/((x^2 +3)^2 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left(\mathrm{2}{x}+\mathrm{1}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 77991    Answers: 2   Comments: 0

If P_1 P_2 P_3 will be taken as point in an Argand diagram representing complex number Z_1 ,Z_2 ,Z_3 and point P_(1 ) ,P_2 ,P_3 is an equalateral triangle.show that (Z_2 −Z_3 )^2 +(Z_3 −Z_1 )^2 +(Z_1 −Z_2 )^2 =0

$${If}\:\:{P}_{\mathrm{1}} \:\:{P}_{\mathrm{2}} \:\:{P}_{\mathrm{3}} \:\:{will}\:{be}\:{taken} \\ $$$${as}\:{point}\:{in}\:{an}\:{Argand} \\ $$$${diagram}\:{representing} \\ $$$${complex}\:{number} \\ $$$${Z}_{\mathrm{1}} ,{Z}_{\mathrm{2}} ,{Z}_{\mathrm{3}} \:\:{and}\:{point} \\ $$$${P}_{\mathrm{1}\:} ,{P}_{\mathrm{2}} ,{P}_{\mathrm{3}} \:{is}\:{an}\:{equalateral} \\ $$$${triangle}.{show}\:{that} \\ $$$$\left({Z}_{\mathrm{2}} −{Z}_{\mathrm{3}} \right)^{\mathrm{2}} +\left({Z}_{\mathrm{3}} −{Z}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({Z}_{\mathrm{1}} −{Z}_{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{0} \\ $$

Question Number 77990    Answers: 2   Comments: 0

Find the equation to the two circles each of which touch the three circle x^2 +y^2 =4a^2 x^2 +y^2 +2ax=0 x^2 +y^2 −2ax=0

$${Find}\:{the}\:{equation}\:{to}\:{the} \\ $$$${two}\:{circles}\:{each}\:{of} \\ $$$${which}\:{touch}\:{the}\:{three}\:{circle} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{4}{a}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{ax}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{ax}=\mathrm{0} \\ $$$$ \\ $$

Question Number 77988    Answers: 1   Comments: 1

Question Number 77987    Answers: 0   Comments: 0

Question Number 77966    Answers: 1   Comments: 0

Question Number 77965    Answers: 1   Comments: 9

solve for x,y,z ∈N 35x+21y+60z=665

$${solve}\:{for}\:{x},{y},{z}\:\in\mathbb{N} \\ $$$$\mathrm{35}{x}+\mathrm{21}{y}+\mathrm{60}{z}=\mathrm{665} \\ $$

Question Number 77962    Answers: 1   Comments: 2

∫(dx/(1+(tan(x))^(√2) )) dx

$$\int\frac{{dx}}{\mathrm{1}+\left({tan}\left({x}\right)\right)^{\sqrt{\mathrm{2}}} }\:{dx} \\ $$

Question Number 77960    Answers: 0   Comments: 3

∫ ((2x^3 −1)/(x^4 +x)) dx?

$$\int\:\frac{\mathrm{2}{x}^{\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{4}} +{x}}\:{dx}? \\ $$

Question Number 77959    Answers: 2   Comments: 0

solve tan ((1/(1+x^2 )))>1

$${solve}\:\mathrm{tan}\:\left(\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)>\mathrm{1}\: \\ $$

Question Number 77953    Answers: 2   Comments: 0

Given that Σ_(r=0) ^4 6r =2Σ_(r=1) ^n 5r, work out the value of n.

$$\mathrm{Given}\:\mathrm{that}\:\underset{\mathrm{r}=\mathrm{0}} {\overset{\mathrm{4}} {\sum}}\mathrm{6r}\:=\mathrm{2}\underset{\mathrm{r}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{5r},\:\mathrm{work}\:\mathrm{out}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{n}. \\ $$

Question Number 78012    Answers: 1   Comments: 3

prove that lim_(x→1) ((sin(𝛑cos𝛑x))/((x−1)^2 )) = − (𝛑^3 /2)

$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{1}} {\boldsymbol{{lim}}}\frac{\boldsymbol{{sin}}\left(\boldsymbol{\pi{cos}\pi{x}}\right)}{\left(\boldsymbol{{x}}−\mathrm{1}\right)^{\mathrm{2}} }\:=\:−\:\frac{\boldsymbol{\pi}^{\mathrm{3}} }{\mathrm{2}} \\ $$

Question Number 77918    Answers: 1   Comments: 1

∫ _0 ^π e^(−2x) sin x dx ?

$$\int\underset{\mathrm{0}} {\overset{\pi} {\:}}\:{e}^{−\mathrm{2}{x}} \:\mathrm{sin}\:{x}\:{dx}\:?\: \\ $$

Question Number 77915    Answers: 1   Comments: 0

Find prime factor of 4^8 − 3^8

$${Find}\:\:{prime}\:\:{factor}\:\:{of}\:\:\mathrm{4}^{\mathrm{8}} \:−\:\mathrm{3}^{\mathrm{8}} \:\: \\ $$

Question Number 77902    Answers: 0   Comments: 8

Question Number 77897    Answers: 1   Comments: 5

calculateU_n = Σ_(k=1) ^n k(−1)^k and v_n =Σ_(k=1) ^n k^2 (−1)^k

$${calculateU}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} \:\:\:{and}\:{v}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} {k}^{\mathrm{2}} \left(−\mathrm{1}\right)^{{k}} \\ $$

Question Number 77891    Answers: 0   Comments: 0

Question Number 77890    Answers: 0   Comments: 0

Question Number 77887    Answers: 1   Comments: 0

∫(e^(sinh(x)) /(cosh(x))) dx

$$\int\frac{{e}^{{sinh}\left({x}\right)} }{{cosh}\left({x}\right)}\:{dx} \\ $$

Question Number 77886    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((arctan(x^2 +x^(−2) ))/(x^2 +a^2 ))dx with a>0 2) find the value of ∫_0 ^∞ ((arctan(x^2 +x^(−2) ))/(x^2 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \:+{x}^{−\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \:+{x}^{−\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 77885    Answers: 2   Comments: 0

solve for : x 1.(√((x−a)(x−b)))+(√((x−b)(x−c)))+(√((x−c)(x−a)))=d [a,b,c,d∈R try for: a=4,b=3,c=2,d=1] 2. (x−a^2 )(√(x−a))+(x−a)(√(x−a^2 ))=a^2 +a+1 3. (x−a^2 )(√(x^2 −a))+(x^2 −a)(√(x−a^2 ))=a^2 +a+1 [a∈R try for: a=(1/2) ]

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\::\:\boldsymbol{\mathrm{x}} \\ $$$$\mathrm{1}.\sqrt{\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}\right)\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{b}}\right)}+\sqrt{\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{b}}\right)\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{c}}\right)}+\sqrt{\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{c}}\right)\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}\right)}=\boldsymbol{\mathrm{d}} \\ $$$$\left[\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}},\boldsymbol{\mathrm{d}}\in\boldsymbol{\mathrm{R}}\right. \\ $$$$\left.\mathrm{try}\:\mathrm{for}:\:\:\mathrm{a}=\mathrm{4},\mathrm{b}=\mathrm{3},\mathrm{c}=\mathrm{2},\mathrm{d}=\mathrm{1}\right] \\ $$$$\mathrm{2}.\:\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} \right)\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}}+\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}\right)\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} }=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{a}}+\mathrm{1} \\ $$$$\mathrm{3}.\:\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} \right)\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}}+\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}\right)\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} }=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{a}}+\mathrm{1} \\ $$$$\left[\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}\right. \\ $$$$\left.\mathrm{try}\:\mathrm{for}:\:\mathrm{a}=\frac{\mathrm{1}}{\mathrm{2}}\:\right] \\ $$$$ \\ $$

Question Number 77883    Answers: 1   Comments: 7

Question Number 77881    Answers: 0   Comments: 5

Question Number 77879    Answers: 0   Comments: 0

∫_1 ^∞ (1/(√(x^3 +5))) dx

$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{3}} +\mathrm{5}}}\:{dx} \\ $$

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