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Question Number 63894 Answers: 0 Comments: 1

sove the (de) x^2 y^′ −(2x+3)y =sin(x^2 ) with y(1)=2 and y^′ (1)=1 .

$${sove}\:{the}\:\left({de}\right)\:{x}^{\mathrm{2}} {y}^{'} \:−\left(\mathrm{2}{x}+\mathrm{3}\right){y}\:={sin}\left({x}^{\mathrm{2}} \right)\:\:{with}\:{y}\left(\mathrm{1}\right)=\mathrm{2}\:{and} \\ $$$${y}^{'} \left(\mathrm{1}\right)=\mathrm{1}\:. \\ $$

Question Number 63893 Answers: 0 Comments: 1

1) simplify W_n (z)=(1+z)(1+z^2 )....(1+z^2^n ) (z from C) 2) simplify P_n (θ) =(1+e^(iθ) )(1+e^(2iθ) ).....(1+e^(i2^n θ) ) and sove P_n (θ)=0

$$\left.\mathrm{1}\right)\:{simplify}\:{W}_{{n}} \left({z}\right)=\left(\mathrm{1}+{z}\right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)....\left(\mathrm{1}+{z}^{\mathrm{2}^{{n}} } \right)\:\left({z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{2}\right)\:{simplify}\:{P}_{{n}} \left(\theta\right)\:=\left(\mathrm{1}+{e}^{{i}\theta} \right)\left(\mathrm{1}+{e}^{\mathrm{2}{i}\theta} \right).....\left(\mathrm{1}+{e}^{{i}\mathrm{2}^{{n}} \theta} \right)\:{and}\:{sove} \\ $$$${P}_{{n}} \left(\theta\right)=\mathrm{0} \\ $$

Question Number 63892 Answers: 0 Comments: 3

calculate A=∫_0 ^∞ (x^(2017) /(1+x^(2019) )) dx and B =∫_0 ^∞ (x^(2019) /(1+x^(2021) )) dx calculate the fraction (A/B)

$${calculate}\:{A}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2017}} }{\mathrm{1}+{x}^{\mathrm{2019}} }\:{dx}\:\:{and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2019}} }{\mathrm{1}+{x}^{\mathrm{2021}} }\:{dx} \\ $$$${calculate}\:{the}\:{fraction}\:\frac{{A}}{{B}} \\ $$

Question Number 63891 Answers: 0 Comments: 0

A bus is travelling along a straight road at 100Km/hr and the bus conductor walks at 6Km/hr on the floor of the bus and in the same direction as the bus. Find the speed of the conductor relative to the road, and relative to the bus. If the bus conductor now works at the same rate but in the opposite direction as the bus, find his new speed relative to the road. Answers in textbook: 106Km/hr, 64Km/hr, 94Km/hr

Question Number 63888 Answers: 2 Comments: 0

y = log_2 [log_3 (log_5 x)] y = ?

$${y}\:=\:{log}_{\mathrm{2}} \left[{log}_{\mathrm{3}} \left({log}_{\mathrm{5}} {x}\right)\right] \\ $$$${y}\:=\:? \\ $$

Question Number 63881 Answers: 0 Comments: 1

∫e^x /Lnxdx

$$\int{e}^{{x}} /{Lnxdx} \\ $$

Question Number 63883 Answers: 0 Comments: 1

∫ln(x)ln(1−x)ln(1−2x)dx

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

Question Number 63865 Answers: 0 Comments: 2

If ∫ ((4 e^x + 6 e^(−x) )/(9 e^x − 4 e^(−x) ))dx=Ax+B log(9e^(2x) −4)+C, then

$$\mathrm{If}\:\int\:\:\frac{\mathrm{4}\:{e}^{{x}} +\:\mathrm{6}\:{e}^{−{x}} }{\mathrm{9}\:{e}^{{x}} −\:\mathrm{4}\:{e}^{−{x}} }{dx}={Ax}+{B}\:\mathrm{log}\left(\mathrm{9}{e}^{\mathrm{2}{x}} −\mathrm{4}\right)+{C}, \\ $$$$\mathrm{then} \\ $$

Question Number 63862 Answers: 0 Comments: 2

If the 3rd term in the expansion of [x+x^(log_(10) x) ]^5 is 10^6 , then x may be

$$\mathrm{If}\:\mathrm{the}\:\mathrm{3rd}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left[{x}+{x}^{\mathrm{log}_{\mathrm{10}} {x}} \right]^{\mathrm{5}} \mathrm{is}\:\mathrm{10}^{\mathrm{6}} ,\:\mathrm{then}\:{x}\:\mathrm{may}\:\mathrm{be} \\ $$

Question Number 63861 Answers: 0 Comments: 2

The number of non−zero terms in the expansion of (1+3(√2) x)^9 +(1−3(√2) x)^9 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}−\mathrm{zero}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{3}\sqrt{\mathrm{2}}\:{x}\right)^{\mathrm{9}} +\left(\mathrm{1}−\mathrm{3}\sqrt{\mathrm{2}}\:{x}\right)^{\mathrm{9}} \:\mathrm{is} \\ $$

Question Number 63860 Answers: 1 Comments: 0

If n is even positive integer, then the condition that the greatest term in the expansion of (1+x)^n may have the greatest coefficient also is

$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{condition}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{n}} \:\mathrm{may}\:\mathrm{have}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{also}\:\mathrm{is} \\ $$

Question Number 63858 Answers: 1 Comments: 0

if a_1 , a_2 , a_3 , a_4 are the coefficient of any four four consecutive terms in the expansion of (1+x)^n then (a_1 /(a_2 +a_1 ))+(a_3 /(a_3 +a_4 )) is equal to...

Question Number 63857 Answers: 0 Comments: 2

Question Number 63852 Answers: 0 Comments: 0

prove that ∫_0 ^1 arctan(x) cot(((πx)/2)) dx = ((3 ln^2 (2))/(2π))+((lnπ ln2)/π)+∫_0 ^∞ ((ln(1+x^2 ))/(e^(2πx) +1)) dx

$${prove}\:{that} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {arctan}\left({x}\right)\:{cot}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:{dx}\:=\:\frac{\mathrm{3}\:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}\pi}+\frac{{ln}\pi\:{ln}\mathrm{2}}{\pi}+\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{e}^{\mathrm{2}\pi{x}} +\mathrm{1}}\:{dx} \\ $$

Question Number 63845 Answers: 0 Comments: 1

Σ_(r=0) ^n ^n C_r ((1+r log_e 10)/((1+ log_e 10^n )^r ))=...

$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{r}} \:\frac{\mathrm{1}+{r}\:\mathrm{log}_{{e}} \:\mathrm{10}}{\left(\mathrm{1}+\:\mathrm{log}_{{e}} \:\mathrm{10}^{{n}} \right)^{{r}} }=... \\ $$

Question Number 63844 Answers: 3 Comments: 3

∫(1+4x+x^2 )^m dx

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

Question Number 63836 Answers: 0 Comments: 1

If n ∈ N, then the sum of the coefficients in the expansion of the binomial (5x−4y)^n is

$$\mathrm{If}\:{n}\:\in\:{N},\:\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\left(\mathrm{5}{x}−\mathrm{4}{y}\right)^{{n}} \:\mathrm{is} \\ $$

Question Number 63835 Answers: 0 Comments: 1

If n ∈ N, then the sum of the coefficients in the expansion of the binomial (5x−4y)^n is

Question Number 63834 Answers: 0 Comments: 2

The 14th term from the end in the expansion of ((√x) − (√y))^(17) is

$$\mathrm{The}\:\mathrm{14th}\:\mathrm{term}\:\mathrm{from}\:\mathrm{the}\:\mathrm{end}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\sqrt{{x}}\:−\:\sqrt{{y}}\right)^{\mathrm{17}} \:\mathrm{is} \\ $$

Question Number 63833 Answers: 0 Comments: 0

If the binomial coefficients of 2nd, 3rd and 4th terms in the expansion of [(√2^(log_(10) (10−3^x )) ) + (2^((x−2) log_(10) 3) )^(1/5) ]^m are in AP and the 6th term is 21, then the value(s) of x is(are)

$$\mathrm{If}\:\mathrm{the}\:\mathrm{binomial}\:\mathrm{coefficients}\:\mathrm{of}\:\mathrm{2nd},\:\mathrm{3rd} \\ $$$$\mathrm{and}\:\mathrm{4th}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left[\sqrt{\mathrm{2}^{\mathrm{log}_{\mathrm{10}} \left(\mathrm{10}−\mathrm{3}^{{x}} \right)} }\:+\:\sqrt[{\mathrm{5}}]{\mathrm{2}^{\left({x}−\mathrm{2}\right)\:\mathrm{log}_{\mathrm{10}} \mathrm{3}} }\right]^{{m}} \:\mathrm{are}\:\mathrm{in} \\ $$$$\mathrm{AP}\:\mathrm{and}\:\mathrm{the}\:\mathrm{6th}\:\mathrm{term}\:\mathrm{is}\:\mathrm{21},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\left(\mathrm{s}\right) \\ $$$$\mathrm{of}\:{x}\:\:\mathrm{is}\left(\mathrm{are}\right) \\ $$

Question Number 63832 Answers: 0 Comments: 5

The largest coefficient in the expansion of (1+x)^(24) is

$$\mathrm{The}\:\mathrm{largest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{24}} \:\mathrm{is} \\ $$

Question Number 63831 Answers: 1 Comments: 0

If C_r be the coefficient of x^r in (1+x)^n , then the value of Σ_(r=0) ^n (r+1)^2 C_r is

$$\mathrm{If}\:{C}_{{r}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\left(\mathrm{1}+{x}\right)^{{n}} , \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left({r}+\mathrm{1}\right)^{\mathrm{2}} \:{C}_{{r}} \:\mathrm{is} \\ $$

Question Number 63824 Answers: 0 Comments: 1

solve y^′ (√(2x−1)) +y(x^2 +3) =xsin(2x)

$${solve}\:{y}^{'} \sqrt{\mathrm{2}{x}−\mathrm{1}}\:+{y}\left({x}^{\mathrm{2}} +\mathrm{3}\right)\:={xsin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 63822 Answers: 0 Comments: 1

find ∫ (x^2 +1)(√((x+1)/(x−2)))dx