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AllQuestion and Answers: Page 1732

Question Number 33032 Answers: 1 Comments: 5

f:N→R f(1)=2005. and f(1)+f(2)+......+f(n)= n^2 f(n),n>1. Then f(2004)=?

$${f}:{N}\rightarrow{R} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2005}. \\ $$$${and}\: \\ $$$${f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+......+{f}\left({n}\right)=\:{n}^{\mathrm{2}} \:{f}\left({n}\right),{n}>\mathrm{1}. \\ $$$${Then}\:{f}\left(\mathrm{2004}\right)=? \\ $$

Question Number 33028 Answers: 0 Comments: 0

find the value of∫_0 ^∞ (e^(−[t]) /(t+1))dt .

$${find}\:{the}\:{value}\:{of}\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\left[{t}\right]} }{{t}+\mathrm{1}}{dt}\:\:. \\ $$

Question Number 33027 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (x^3 /(1+x^5 ))dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{5}} }{dx}. \\ $$

Question Number 33026 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((1+x^4 )/(1+x^6 )) dx .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\:{dx}\:. \\ $$

Question Number 33009 Answers: 2 Comments: 1

help ! ! ! ∫ (dx/(csc(x)−1)) = ? [ my way ] ∫( (dx/((1/(sinx)) − 1)) ) =∫((sinx)/(1−sinx)) dx =−∫ ((sinx−1+1)/(sinx−1)) dx =−∫1+(1/(sinx−1)) dx =−(∫1dx+∫((sinx+1)/((sinx−1)(sinx+1))) dx) =−(x+C−∫((sinx+1)/(1−sin^2 x)) dx) =−(x+C−∫ ((sinx)/(cos^2 x)) dx−∫ (1/(cos^2 x)) dx) =−(x+C+∫(cosx)^(−2) dcosx−∫(1/(cos^2 x))dx) =−(x−(cosx)^(−1) +C−∫(1/(cos^2 x))dx) ...and I can′t solve the ∫(1/(cos^2 x))dx oh i just found that is tanx+C

$${help}\:!\:!\:! \\ $$$$\int\:\frac{{dx}}{{csc}\left({x}\right)−\mathrm{1}}\:=\:? \\ $$$$ \\ $$$$\left[\:{my}\:{way}\:\right] \\ $$$$\int\left(\:\frac{{dx}}{\frac{\mathrm{1}}{{sinx}}\:−\:\mathrm{1}}\:\right) \\ $$$$=\int\frac{{sinx}}{\mathrm{1}−{sinx}}\:{dx} \\ $$$$=−\int\:\frac{{sinx}−\mathrm{1}+\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\ $$$$=−\int\mathrm{1}+\frac{\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\ $$$$=−\left(\int\mathrm{1}{dx}+\int\frac{{sinx}+\mathrm{1}}{\left({sinx}−\mathrm{1}\right)\left({sinx}+\mathrm{1}\right)}\:{dx}\right) \\ $$$$=−\left({x}+{C}−\int\frac{{sinx}+\mathrm{1}}{\mathrm{1}−{sin}^{\mathrm{2}} {x}}\:{dx}\right) \\ $$$$=−\left({x}+{C}−\int\:\frac{{sinx}}{{cos}^{\mathrm{2}} {x}}\:{dx}−\int\:\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}\:{dx}\right) \\ $$$$=−\left({x}+{C}+\int\left({cosx}\right)^{−\mathrm{2}} {dcosx}−\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx}\right) \\ $$$$=−\left({x}−\left({cosx}\right)^{−\mathrm{1}} +{C}−\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx}\right) \\ $$$$...{and}\:{I}\:{can}'{t}\:{solve}\:{the}\:\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx} \\ $$$$ \\ $$$${oh}\:{i}\:{just}\:{found}\:{that}\:{is}\:{tanx}+{C} \\ $$

Question Number 33005 Answers: 0 Comments: 1

Given that y= ((sin x)/(1 + cos x)) find (dy/dx) Evaluate ∫_1 ^2 (x + 4)dx

$$\:{Given}\:{that}\: \\ $$$$\:\:\:{y}=\:\frac{{sin}\:{x}}{\mathrm{1}\:+\:{cos}\:{x}}\:{find}\:\frac{{dy}}{{dx}} \\ $$$${Evaluate}\: \\ $$$$\:\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \left({x}\:+\:\mathrm{4}\right){dx} \\ $$

Question Number 32999 Answers: 1 Comments: 1

Question Number 32998 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ ((4n+1)/(n^2 (3n+1)^2 )) .

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{4}{n}+\mathrm{1}}{{n}^{\mathrm{2}} \left(\mathrm{3}{n}+\mathrm{1}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 32997 Answers: 0 Comments: 1

calculate Σ_(n=0) ^∞ ((2n+3)/((n+1)^2 (n+2)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{2}{n}+\mathrm{3}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

Question Number 32996 Answers: 0 Comments: 1

find the sequence (v_n ) wich verify v_(n+2) =(√(v_n .v_(n+1) )) .

$${find}\:{the}\:{sequence}\:\left({v}_{{n}} \right)\:{wich}\:{verify}\:\:{v}_{{n}+\mathrm{2}} \:=\sqrt{{v}_{{n}} \:.{v}_{{n}+\mathrm{1}} }\:. \\ $$

Question Number 32995 Answers: 0 Comments: 1

let u_0 =a , u_1 =b and u_(n+2) =(1/2)(u_n +u_(n+1) ) 1) find u_n interms of n 2) find lim_(n→∞) u_n if a=0

$${let}\:{u}_{\mathrm{0}} ={a}\:,\:{u}_{\mathrm{1}} ={b}\:{and}\:{u}_{{n}+\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} \:+{u}_{{n}+\mathrm{1}} \right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{u}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} \:\:{if}\:{a}=\mathrm{0} \\ $$$$ \\ $$

Question Number 32994 Answers: 0 Comments: 1

find ∫_0 ^∞ ((1−cos(λx))/x^2 ) dx with λ>0 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}−{cos}\left(\lambda{x}\right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32993 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−λx) ((sinx)/(√x)) dx wih λ>0 .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{x}} \:\frac{{sinx}}{\sqrt{{x}}}\:{dx}\:\:{wih}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32992 Answers: 0 Comments: 0

find lim_(x→0) ((tan(sinx) −sin(tanx))/x^2 ) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{tan}\left({sinx}\right)\:−{sin}\left({tanx}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32991 Answers: 1 Comments: 1

calculate lim_(x→0) ((ln(1+x) −x)/x^2 ) .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)\:−{x}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32990 Answers: 1 Comments: 1

find lim_(x→0^+ ) x e^(−lnx) .

$$\:\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{x}\:{e}^{−{lnx}} \:\:. \\ $$

Question Number 32989 Answers: 0 Comments: 0

let give f(x)=(((√(2+x)) −(√(2−x)) )/x) find f^(−1) (x) and calculate (f^(−1) (x))^′ .

$${let}\:{give}\:{f}\left({x}\right)=\frac{\sqrt{\mathrm{2}+{x}}\:−\sqrt{\mathrm{2}−{x}}\:}{{x}} \\ $$$${find}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{calculate}\:\left({f}^{−\mathrm{1}} \left({x}\right)\right)^{'} \:\:. \\ $$

Question Number 32987 Answers: 1 Comments: 0

If 1=1 2=4 3=10 4=20 5=?

$${If}\:\mathrm{1}=\mathrm{1} \\ $$$$\mathrm{2}=\mathrm{4} \\ $$$$\mathrm{3}=\mathrm{10}\: \\ $$$$\mathrm{4}=\mathrm{20} \\ $$$$\mathrm{5}=? \\ $$

Question Number 32985 Answers: 1 Comments: 0

Question Number 32980 Answers: 1 Comments: 0

(d/dx)( determinant ((((x^2 +2)^(x^3 +3) ),2^x ,(cosx^x )),((log_2^(x+1) (x^2 +x^(2x) +3^ )),(xlnx),(sin^(−1) tanx)),(3,π^(sinhx) ,e^x )))=?

$$ \\ $$$$ \\ $$$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\begin{vmatrix}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{x}^{\mathrm{3}} +\mathrm{3}} }&{\mathrm{2}^{\mathrm{x}} }&{\mathrm{cosx}^{\mathrm{x}} }\\{\mathrm{log}_{\mathrm{2}^{\mathrm{x}+\mathrm{1}} } \left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2x}} +\overset{} {\mathrm{3}}\right)}&{\mathrm{xlnx}}&{\mathrm{sin}^{−\mathrm{1}} \mathrm{tanx}}\\{\mathrm{3}}&{\pi^{\mathrm{sinhx}} }&{\mathrm{e}^{\mathrm{x}} }\end{vmatrix}\right)=? \\ $$$$ \\ $$

Question Number 33152 Answers: 0 Comments: 1

it is given that (1/n)Σ_(r=1) ^n x^r =2 and (√((1/n)Σ_(r=1) ^n (x_r )^2 −(1/n^2 )(Σ_(r=1) ^n )^2 ))= 3 determine in terms of n the value of. Σ_(r=1) ^n (x_r +1)^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}^{{r}} =\mathrm{2}\:{and}\:\sqrt{\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} \right)^{\mathrm{2}} −\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\right)^{\mathrm{2}} }=\:\mathrm{3} \\ $$$${determine}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value} \\ $$$${of}. \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left({x}_{{r}} +\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 32977 Answers: 1 Comments: 0

Prove that ^n C_r +^n C_(r+1) =^(n+1) C_(r+1)

$${Prove}\:{that}\:\:^{{n}} {C}_{{r}} \:\:+\:^{{n}} {C}_{{r}+\mathrm{1}} \:=\:^{{n}+\mathrm{1}} {C}_{{r}+\mathrm{1}} \\ $$$$ \\ $$

Question Number 33125 Answers: 0 Comments: 0

let give u_n = ∫_0 ^π ((cos(nx)dx)/(1−2λcosx +λ^2 )) 1) prove that λ u_(n+2) −(1+λ^2 )u_(n+1) +λ u_n =0 2) ptove that Σ u_n is convergent and find its sum

$${let}\:{give}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{cos}\left({nx}\right){dx}}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\lambda\:{u}_{{n}+\mathrm{2}} \:−\left(\mathrm{1}+\lambda^{\mathrm{2}} \right){u}_{{n}+\mathrm{1}} \:+\lambda\:{u}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{ptove}\:{that}\:\Sigma\:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum} \\ $$

Question Number 33124 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ (1/(n(n+1)2^(n−1) )) .

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\mathrm{2}^{{n}−\mathrm{1}} }\:. \\ $$

Question Number 32971 Answers: 1 Comments: 0

If siny=xsin(a+y) show that (dy/dx)=((sina)/(1−2xcosa+x^2 ))

$${If}\:{siny}={xsin}\left({a}+{y}\right)\:{show}\:{that} \\ $$$$\frac{{dy}}{{dx}}=\frac{{sina}}{\mathrm{1}−\mathrm{2}{xcosa}+{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 32968 Answers: 1 Comments: 0

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