Question and Answers Forum

AllQuestion and Answers: Page 1565

Question Number 42771 Answers: 0 Comments: 0

1) find ∫_0 ^1 ((ln(x))/(1−x^2 ))dx 2) find ∫_0 ^1 ((ln(x))/(1−x^4 ))dx

$$\left.\mathrm{1}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}\right)}{\mathrm{1}−{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 42770 Answers: 0 Comments: 0

1)find A(ξ) = ∫_0 ^ξ ln(x)ln(1−x)dx with 0<ξ<1 2) calculate ∫_0 ^1 ln(x)ln(1−x)dx

$$\left.\mathrm{1}\right){find}\:{A}\left(\xi\right)\:=\:\int_{\mathrm{0}} ^{\xi} {ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){dx}\:\:{with}\:\:\mathrm{0}<\xi<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){dx} \\ $$

Question Number 42769 Answers: 0 Comments: 0

find ∫_0 ^1 (x^2 /(1+xe^(−x) )) dx .

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{xe}^{−{x}} }\:{dx}\:. \\ $$

Question Number 42768 Answers: 0 Comments: 0

Question Number 42766 Answers: 0 Comments: 0

Question Number 42765 Answers: 0 Comments: 1

Question Number 42763 Answers: 0 Comments: 1

For A = {1, 2, 3}, let B be the set of 2−element sets belonging to P(A) and let C be the set consisting of the sets that are intersections of two distinct elements of B. Determine C P(A) = power set of A

$$\mathrm{For}\:{A}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\},\:\mathrm{let}\:{B}\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{2}−\mathrm{element}\:\mathrm{sets} \\ $$$$\mathrm{belonging}\:\mathrm{to}\:{P}\left({A}\right)\:\mathrm{and}\:\mathrm{let}\:{C}\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{consisting}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sets}\:\mathrm{that}\:\mathrm{are}\:\mathrm{intersections}\:\mathrm{of}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{elements} \\ $$$$\mathrm{of}\:{B}.\:\mathrm{Determine}\:{C} \\ $$$$ \\ $$$${P}\left({A}\right)\:=\:\mathrm{power}\:\mathrm{set}\:\mathrm{of}\:{A} \\ $$

Question Number 42761 Answers: 0 Comments: 0

Question Number 42758 Answers: 0 Comments: 0

Question Number 42756 Answers: 0 Comments: 1

33(√(67))

$$\mathrm{33}\sqrt{\mathrm{67}} \\ $$

Question Number 42730 Answers: 1 Comments: 1

Given that f(x) = (√(1−x)) Find a) D_f for the arranged form of f(x) b) fg if fh= g(x) and h(x)= 3x^2 −4 c) A(x)= { (((√(1−x)) , x≠ 1)),((x^2 ,x≠0)) :} find A^(−1) .

$${Given}\:{that}\:{f}\left({x}\right)\:=\:\sqrt{\mathrm{1}−{x}}\:{Find} \\ $$$$\left.{a}\right)\:{D}_{{f}} \:{for}\:{the}\:{arranged}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.{b}\right)\:{fg}\:{if}\:{fh}=\:{g}\left({x}\right)\:{and}\:{h}\left({x}\right)=\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4} \\ $$$$\left.{c}\right)\:{A}\left({x}\right)=\:\begin{cases}{\sqrt{\mathrm{1}−{x}}\:,\:{x}\neq\:\mathrm{1}}\\{{x}^{\mathrm{2}} ,{x}\neq\mathrm{0}}\end{cases} \\ $$$${find}\:{A}^{−\mathrm{1}} . \\ $$$$ \\ $$$$ \\ $$

Question Number 42728 Answers: 2 Comments: 0

If A= [(( cos x),(sin x)),((−sin x),(cos x)) ] and A adj A = k [(1,0),(0,1) ], then the value of k is

$$\mathrm{If}\:{A}=\begin{bmatrix}{\:\:\:\mathrm{cos}\:{x}}&{\mathrm{sin}\:{x}}\\{−\mathrm{sin}\:{x}}&{\mathrm{cos}\:{x}}\end{bmatrix}\:\mathrm{and}\: \\ $$$${A}\:\mathrm{adj}\:{A}\:=\:{k}\begin{bmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is} \\ $$

Question Number 42725 Answers: 1 Comments: 0

Question Number 42719 Answers: 0 Comments: 1

Question Number 42718 Answers: 0 Comments: 0

if p(A)=0.25 and p(B)=0.8 then show that 0.05≤p(A∩B)≤0.25

$$\mathrm{if}\:\mathrm{p}\left(\mathrm{A}\right)=\mathrm{0}.\mathrm{25}\:\mathrm{and}\:\mathrm{p}\left(\mathrm{B}\right)=\mathrm{0}.\mathrm{8} \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\mathrm{0}.\mathrm{05}\leqslant\mathrm{p}\left(\mathrm{A}\cap\mathrm{B}\right)\leqslant\mathrm{0}.\mathrm{25} \\ $$

Question Number 42713 Answers: 1 Comments: 1

Question Number 42711 Answers: 1 Comments: 0

If 2 sides of a triangle are i^ +2j^ and i^ +k^ , then find all possible third side ?

$$\mathrm{If}\:\mathrm{2}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{are}\:\hat {\mathrm{i}}+\mathrm{2}\hat {\mathrm{j}}\:\mathrm{and} \\ $$$$\hat {\mathrm{i}}+\hat {\mathrm{k}}\:,\:\mathrm{then}\:\mathrm{find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{third}\:\mathrm{side}\:? \\ $$

Question Number 42709 Answers: 1 Comments: 3

If f(x)= x^3 −((3x^2 )/2) +x + (1/4). Then ∫_(1/4) ^(3/4) f(f(x))dx =?

$$\mathrm{If}\:\mathrm{f}\left({x}\right)=\:{x}^{\mathrm{3}} \:−\frac{\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}}\:+{x}\:+\:\frac{\mathrm{1}}{\mathrm{4}}. \\ $$$${T}\mathrm{hen}\:\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\frac{\mathrm{3}}{\mathrm{4}}} \:\mathrm{f}\left(\mathrm{f}\left({x}\right)\right)\mathrm{d}{x}\:=? \\ $$

Question Number 42704 Answers: 0 Comments: 4

f(x) = ((e^(3x) +e^(−3x) )/2) 1) determine f^(−1) (x) 2) calculate ∫_0 ^1 x f(x)dx and ∫_0 ^1 f(x)dx 3) calculate ∫ f^(−1) (x)dx 4) calculate u_n = ∫_0 ^π f(x)cos(nx)dx and v_n = ∫_0 ^n f(x)sin(nx)dx find nature of Σ (v_n /u_n ) ∫_0 ^1 xf(x) dx =(1/2) ∫_0 ^1 x e^(3x) dx +(1/2) ∫_0 ^1 x e^(−3x) dx (by parts) =(1/2){ [(x/3)e^(3x) ]_0 ^1 −(1/3)∫_0 ^1 e^(3x) dx +[−(x/3)e^(−3x) ]_0 ^1 +(1/3)∫_0 ^1 e^(−3x) dx} =(1/2){(e^3 /3) −(1/9)(e^3 −1) −(e^(−3) /3) −(1/9)(e^(−3) −1)}

$${f}\left({x}\right)\:\:=\:\:\frac{{e}^{\mathrm{3}{x}} \:+{e}^{−\mathrm{3}{x}} }{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}\:{f}\left({x}\right){dx}\:\:\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\:\:\int\:\:{f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:{f}\left({x}\right){cos}\left({nx}\right){dx}\:{and}\:{v}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{f}\left({x}\right){sin}\left({nx}\right){dx} \\ $$$${find}\:{nature}\:{of}\:\Sigma\:\frac{{v}_{{n}} }{{u}_{{n}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{xf}\left({x}\right)\:{dx}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{\mathrm{3}{x}} {dx}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{−\mathrm{3}{x}} {dx}\:\:\:\left({by}\:{parts}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\left[\frac{{x}}{\mathrm{3}}{e}^{\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:−\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{\mathrm{3}{x}} {dx}\:\:+\left[−\frac{{x}}{\mathrm{3}}{e}^{−\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:+\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−\mathrm{3}{x}} {dx}\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{{e}^{\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{\mathrm{3}} −\mathrm{1}\right)\:−\frac{{e}^{−\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{−\mathrm{3}} −\mathrm{1}\right)\right\} \\ $$$$ \\ $$

Question Number 42695 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (x^4 /(x^(8 ) +16))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{8}\:} \:+\mathrm{16}}{dx} \\ $$

Question Number 42689 Answers: 0 Comments: 3

let f(x) = (x/(x^3 −2x +1)) 1) find D_f 2) find f^((n)) (x) then f^((n)) (0) 3) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:=\:\frac{{x}}{{x}^{\mathrm{3}} −\mathrm{2}{x}\:\:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{then}\:\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 42688 Answers: 0 Comments: 3

let g(x) =((x−1)/(x^2 +x +1)) 1) find g^((n)) (x) 2)calculate g^((n)) (0) 3) developp g at integr serie.

$${let}\:{g}\left({x}\right)\:=\frac{{x}−\mathrm{1}}{{x}^{\mathrm{2}} +{x}\:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{g}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{g}\:{at}\:\:{integr}\:{serie}. \\ $$

Question Number 42684 Answers: 0 Comments: 2

The coefficient of x^4 in the expansion of ((x/2) − (3/x^2 ))^(10) is