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Question Number 53252 Answers: 3 Comments: 1

Question Number 53261 Answers: 0 Comments: 0

1)find f(x)=∫_0 ^1 e^(−2t) ln(1−xt)dt with ∣x∣<1 2) calculate ∫_0 ^1 e^(−2t) ln(1−((t(√2))/2))dt.

$$\left.\mathrm{1}\right){find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\mathrm{2}{t}} {ln}\left(\mathrm{1}−{xt}\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\mathrm{2}{t}} {ln}\left(\mathrm{1}−\frac{{t}\sqrt{\mathrm{2}}}{\mathrm{2}}\right){dt}. \\ $$

Question Number 53259 Answers: 1 Comments: 0

Question Number 53228 Answers: 0 Comments: 3

1) find f(a) =∫_0 ^1 (dx/((ax+1)(√(x^2 −x+1)))) with a>0 2) calculate f^′ (a) 3)find the value of ∫_0 ^1 ((xdx)/((ax+1)^2 (√(x^2 −x+1)))) 4) calculate ∫_0 ^1 (dx/((2x+1)(√(x^2 −x+1)))) and ∫_0 ^1 ((xdx)/((2x+1)^2 (√(x^2 −x+1))))

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\left({ax}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\:\:\:{with}\:\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xdx}}{\left({ax}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$

Question Number 53257 Answers: 0 Comments: 1

Question Number 53212 Answers: 2 Comments: 21

Let f(x) = ((2x)/(x^2 + 4)) (a) Find ∫_(−b) ^b f(x) dx, for b > 0 (b) Determine ∫_(−∞) ^∞ f(x) dx is convergent or not

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+\:\mathrm{4}} \\ $$$$ \\ $$$$\left({a}\right)\:\mathrm{Find}\:\underset{−{b}} {\overset{{b}} {\int}}\:{f}\left({x}\right)\:{dx},\:\mathrm{for}\:{b}\:>\:\mathrm{0} \\ $$$$\left({b}\right)\:\mathrm{Determine}\:\underset{−\infty} {\overset{\infty} {\int}}\:{f}\left({x}\right)\:{dx}\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{not} \\ $$

Question Number 53210 Answers: 1 Comments: 0

Question Number 53207 Answers: 0 Comments: 0

Question Number 53205 Answers: 1 Comments: 1

Question Number 53188 Answers: 1 Comments: 0

With a constant throttle setting a motor boat travels 36 miles downstream and then returns. The downstream takes 6 hours less than the return trip. when the speed of the motor boat is doubled, the trip downstream is 1 hour less than the return trip. what is the rate of the streams current? help please sir

$${With}\:{a}\:{constant}\:{throttle}\:{setting}\: \\ $$$${a}\:{motor}\:{boat}\:{travels}\:\mathrm{36}\:{miles} \\ $$$${downstream}\:{and}\:{then}\:{returns}.\:{The} \\ $$$${downstream}\:{takes}\:\mathrm{6}\:{hours}\:{less}\:{than}\: \\ $$$${the}\:{return}\:{trip}.\:{when}\:{the}\:{speed}\:{of}\: \\ $$$${the}\:{motor}\:{boat}\:{is}\:{doubled},\:{the}\:{trip} \\ $$$${downstream}\:{is}\:\mathrm{1}\:{hour}\:{less}\:{than}\:{the}\: \\ $$$${return}\:{trip}.\:{what}\:{is}\:{the}\:{rate}\:{of}\:{the} \\ $$$${streams}\:{current}? \\ $$$${help}\:{please}\:{sir} \\ $$

Question Number 53168 Answers: 2 Comments: 6

Question Number 53165 Answers: 2 Comments: 0

Question Number 53161 Answers: 3 Comments: 0

Question Number 53156 Answers: 2 Comments: 0

Question Number 53152 Answers: 0 Comments: 0

A disc of radius r suspended from a point lie on itself.Find out the minimum time period of oscillation of the disc.

$${A}\:{disc}\:{of}\:{radius}\:{r}\:{suspended}\:{from} \\ $$$${a}\:{point}\:{lie}\:{on}\:{itself}.{Find}\:{out}\:{the} \\ $$$${minimum}\:{time}\:{period}\:{of}\:{oscillation} \\ $$$${of}\:{the}\:{disc}. \\ $$

Question Number 53151 Answers: 1 Comments: 0

A disc of radius r suspended from a point lie on itself.Find out the minimum time period of oscillation of the disc.

Question Number 53145 Answers: 2 Comments: 2

Question Number 53144 Answers: 2 Comments: 1

Find all integers x and y such that ((xy)/(x+y)) is also integer.

$${Find}\:{all}\:{integers}\:{x}\:{and}\:{y}\:{such}\:{that} \\ $$$$\frac{{xy}}{{x}+{y}}\:{is}\:{also}\:{integer}. \\ $$

Question Number 53141 Answers: 1 Comments: 0

If n is an integer greater than unity, then the value of a−^n C_1 (a−1)+^n C_2 (a−2)−^n C_3 (a−3)+... ..+(−1)^n (a−n) is

$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{unity},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${a}−\:^{{n}} {C}_{\mathrm{1}} \left({a}−\mathrm{1}\right)+\:^{{n}} {C}_{\mathrm{2}} \left({a}−\mathrm{2}\right)−\:^{{n}} {C}_{\mathrm{3}} \left({a}−\mathrm{3}\right)+... \\ $$$$\:\:\:\:\:\:\:..+\left(−\mathrm{1}\right)^{{n}} \left({a}−{n}\right)\:\mathrm{is} \\ $$

Question Number 53119 Answers: 6 Comments: 3

Evaluate : 1) ∫(√((2−x)/(4+x))) dx 2) ∫ (√((x−2)/(x−4))) dx 3) ∫ (√((x−2)(x−4))) dx 4) ∫ (dx/(2sinx+3secx)) .

$${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right)\:\int\sqrt{\frac{\mathrm{2}−{x}}{\mathrm{4}+{x}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:\int\:\sqrt{\frac{{x}−\mathrm{2}}{{x}−\mathrm{4}}}\:{dx} \\ $$$$\left.\mathrm{3}\right)\:\int\:\sqrt{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{4}\right)}\:{dx} \\ $$$$\left.\mathrm{4}\right)\:\int\:\frac{{dx}}{\mathrm{2sin}\boldsymbol{{x}}+\mathrm{3sec}\boldsymbol{{x}}}\:. \\ $$

Question Number 53118 Answers: 1 Comments: 0

If a<∫_0 ^(2π) (1/(10+3 cos x)) dx<b, then the ordered pair (a, b) is

$$\mathrm{If}\:{a}<\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{1}}{\mathrm{10}+\mathrm{3}\:\mathrm{cos}\:{x}}\:{dx}<{b},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{ordered}\:\mathrm{pair}\:\left({a},\:{b}\right)\:\mathrm{is} \\ $$

Question Number 53114 Answers: 0 Comments: 1

let A_n =∫_0 ^∞ ((x sin(nx))/((x^2 +n^2 )^2 ))dx with n integr natural not 0 1) find the value of A_n 2) study the convergence of Σ A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}\:{sin}\left({nx}\right)}{\left({x}^{\mathrm{2}} \:+{n}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 53113 Answers: 0 Comments: 1

let I =∫_(−∞) ^(+∞) ((t+1)/((t^2 −t+1)^2 ))dt find value of I .

$${let}\:{I}\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{t}+\mathrm{1}}{\left({t}^{\mathrm{2}} −{t}+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$${find}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 53112 Answers: 1 Comments: 0

calculate ∫_0 ^π ((1+2sinx)/(3 +2cosx))dx let A =∫_0 ^π ((1+2sinx)/(3 +2cosx))dx changement tan((x/2))=t give A =∫_0 ^∞ ((1+((4t)/(1+t^2 )))/(3+2((1−t^2 )/(1+t^2 )))) ((2dt)/(1+t^2 )) =2 ∫_0 ^∞ ((1+t^2 +4t)/((1+t^2 )^2 (((3+3t^2 +2−2t^2 )/(1+t^2 )))))dt =2 ∫_0 ^∞ ((t^2 +4t +1)/((1+t^2 )(5+t^2 )))dt let decompose F(t)=((t^2 +4t+1)/((t^2 +1)(t^2 +5))) F(t)=((at +b)/(t^2 +1)) +((ct +d)/(t^2 +5)) ⇒(at+b)(t^2 +5)+(ct+d)(t^2 +1) =t^2 +4t +1 ⇒ at^3 +5at +bt^2 +5b +ct^3 +ct +dt^2 +d =t^2 +4t +1 ⇒ (a+c)t^3 +(b+d)t^2 +(5a+c)t +5b +d =t^2 +4t +1 ⇒a+c=0 and b+d=1 and 5a+c =4 and 5b+d =1 ⇒c=−a ⇒a=1 ⇒c=−1 we have d=1−b ⇒5b +1−b =1 ⇒b=0 ⇒d=1 ⇒ F(t)=(t/(t^2 +1)) +((−t +1)/(t^2 +5)) ⇒ A =2 ∫_0 ^∞ F(t)dt =∫_0 ^∞ ((2t)/(t^2 +1))dt +∫_0 ^∞ ((−2t +2)/(t^2 +5))dt =[ln(((t^2 +1)/(t^2 +5)))]_0 ^(+∞) +2 ∫_0 ^∞ (dt/(t^2 +5)) =ln(5) + 2 ∫_0 ^∞ (dt/(t^2 +5)) but ∫_0 ^∞ (dt/(t^2 +5))dt =_(t =(√5)u ) ∫_0 ^∞ (((√5)du)/(5(1+u^2 ))) =(1/(√5)) [artanu]_0 ^(+∞) =(π/(2(√5))) ⇒ A =ln(5) +(π/(2(√5))) .

Question Number 53108 Answers: 1 Comments: 0

((x + 2))^(1/3) − ((x − 3))^(1/3) > (1/2)

$$\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:+\:\mathrm{2}}\:\:−\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:−\:\mathrm{3}}\:\:\:>\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 53086 Answers: 0 Comments: 1

JEE Main chemistry errors by Resonance. Today is the last day for raising objectiions. You need to tap once to select and tap the text again to open hyperlink.

$$\mathrm{JEE}\:\mathrm{Main}\:\mathrm{chemistry}\:\mathrm{errors}\:\mathrm{by} \\ $$$$\mathrm{Resonance}.\:\mathrm{Today}\:\mathrm{is}\:\mathrm{the}\:\mathrm{last}\:\mathrm{day} \\ $$$$\mathrm{for}\:\mathrm{raising}\:\mathrm{objectiions}. \\ $$$$\mathrm{You}\:\mathrm{need}\:\mathrm{to}\:\mathrm{tap}\:\mathrm{once}\:\mathrm{to}\:\mathrm{select}\:\mathrm{and}\:\mathrm{tap} \\ $$$$\mathrm{the}\:\mathrm{text}\:\mathrm{again}\:\mathrm{to}\:\mathrm{open}\:\mathrm{hyperlink}. \\ $$

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