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IntegrationQuestion and Answers: Page 167

Question Number 93038 Answers: 1 Comments: 1

∫_1 ^5 (√(x^3 +1)) dx = ?

$$\underset{\mathrm{1}} {\overset{\mathrm{5}} {\int}}\:\sqrt{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 93031 Answers: 1 Comments: 11

Question Number 93005 Answers: 1 Comments: 0

∫ sec x (√(sec x+tan x)) dx =

$$\int\:\mathrm{sec}\:\mathrm{x}\:\sqrt{\mathrm{sec}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\:=\: \\ $$

Question Number 92993 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((cos(cosx−sinx))/(x^2 +4))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{{cos}\left({cosx}−{sinx}\right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 93004 Answers: 0 Comments: 0

∫ ((sin^(−1) (sin (x)))/(sin^4 (x)+cos^2 (x))) dx ?

$$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:\left(\mathrm{x}\right)\right)}{\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}\right)+\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:\mathrm{dx}\:? \\ $$

Question Number 92976 Answers: 1 Comments: 0

define Δ_n =((n(1+n))/2) find a closer form for Σ_(n=1) ^∞ (1/Δ_n ^m )

$${define} \\ $$$$\Delta_{{n}} =\frac{{n}\left(\mathrm{1}+{n}\right)}{\mathrm{2}} \\ $$$${find}\:{a}\:{closer}\:{form}\:{for} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\Delta_{{n}} ^{{m}} } \\ $$

Question Number 92971 Answers: 0 Comments: 3

1) decompose F(x) =(1/(x^4 (x−3)^5 )) 2)calculate ∫_5 ^(+∞) (dx/(x^4 (x−3)^5 ))

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 92960 Answers: 2 Comments: 0

∫ ((sin^2 x dx)/((1+cos x)^2 ))

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

Question Number 92952 Answers: 0 Comments: 1

∫_(−2) ^2 x^3 (cos((x/2))+(1/2))(√(4−x^2 )) dx

$$\int_{−\mathrm{2}} ^{\mathrm{2}} {x}^{\mathrm{3}} \left({cos}\left(\frac{{x}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 92949 Answers: 1 Comments: 2

∫(1/x)sin((1/x)) dx

$$\int\frac{\mathrm{1}}{\mathrm{x}}\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:\mathrm{dx} \\ $$

Question Number 92940 Answers: 0 Comments: 0

prove that Γ(x).Γ(1−x) =(π/(sin(πx))) with 0<x<1

$${prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:{with}\:\mathrm{0}<{x}<\mathrm{1} \\ $$

Question Number 92939 Answers: 0 Comments: 0

find ∫_0 ^1 ((xlnx)/((x^2 +1)^2 ))dx

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{xlnx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 92938 Answers: 1 Comments: 0

find ∫ (dx/((1+cosx)(3−sin^2 x)))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{cosx}\right)\left(\mathrm{3}−{sin}^{\mathrm{2}} {x}\right)} \\ $$

Question Number 92937 Answers: 2 Comments: 1

find ∫_0 ^(2π) (dx/(3cosx +2))

$${find}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{dx}}{\mathrm{3}{cosx}\:+\mathrm{2}} \\ $$

Question Number 92936 Answers: 2 Comments: 2

find ∫_0 ^(2π) (dx/(cosx +sinx))

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dx}}{{cosx}\:+{sinx}} \\ $$

Question Number 92935 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (((x^2 −3)dx)/((x^2 −x+1)^2 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left({x}^{\mathrm{2}} −\mathrm{3}\right){dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 92934 Answers: 0 Comments: 0

calculate ∫_0 ^1 ((xlnx)/((x+1)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{xlnx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 92925 Answers: 0 Comments: 4

S_n =Σ_(k=1) ^∞ (1/((4k^2 −1)^n )) find a simpler form

$${S}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{4}{k}^{\mathrm{2}} −\mathrm{1}\right)^{{n}} } \\ $$$${find}\:{a}\:{simpler}\:{form} \\ $$

Question Number 92889 Answers: 0 Comments: 1

learning distancing ∫ ln((√(1+x))+(√(1−x))) dx

$$\mathrm{learning}\:\mathrm{distancing} \\ $$$$\int\:\mathrm{ln}\left(\sqrt{\mathrm{1}+\mathrm{x}}+\sqrt{\mathrm{1}−\mathrm{x}}\right)\:\mathrm{dx} \\ $$

Question Number 92869 Answers: 0 Comments: 7

∫((−t^3 +2t−t+1)/(t(t^2 +1)))dt

$$\int\frac{−{t}^{\mathrm{3}} +\mathrm{2}{t}−{t}+\mathrm{1}}{{t}\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{dt} \\ $$

Question Number 92804 Answers: 1 Comments: 12

Solve the following differential equations: (i). e^(x−y) dx +e^(y−x) dy=0 (ii). (dy/dx) = (√(y−x)) (iii). (dy/dx)= ((3xy+y^2 )/(3x^2 ))

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{differential}\:\mathrm{equations}: \\ $$$$\:\left(\mathrm{i}\right).\:\mathrm{e}^{\mathrm{x}−\mathrm{y}} \:\mathrm{dx}\:+\mathrm{e}^{\mathrm{y}−\mathrm{x}} \:\mathrm{dy}=\mathrm{0} \\ $$$$\:\:\left(\mathrm{ii}\right).\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\sqrt{\mathrm{y}−\mathrm{x}} \\ $$$$\:\left(\mathrm{iii}\right).\:\frac{\mathrm{dy}}{\mathrm{dx}}=\:\frac{\mathrm{3xy}+\mathrm{y}^{\mathrm{2}} }{\mathrm{3x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 92805 Answers: 1 Comments: 0

Evaluate: ∫_R ∫ ((xy)/(√(1−y^2 ))) dx dy where the region of integration is the positive quadrant of the circle x^2 +y^2 =1.

$$\:\mathrm{Evaluate}: \\ $$$$\:\int_{\boldsymbol{\mathrm{R}}} \int\:\frac{\boldsymbol{\mathrm{xy}}}{\sqrt{\mathrm{1}−\boldsymbol{\mathrm{y}}^{\mathrm{2}} }}\:\boldsymbol{\mathrm{dx}}\:\boldsymbol{\mathrm{dy}}\:\mathrm{where}\:\mathrm{the}\:\mathrm{region}\:\mathrm{of}\:\mathrm{integration}\:\mathrm{is}\:\mathrm{the} \\ $$$$\:\mathrm{positive}\:\mathrm{quadrant}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{1}. \\ $$$$ \\ $$

Question Number 92801 Answers: 0 Comments: 0

Integrate following : (i).∫ (( dx)/(sin x( 3+2cos x))) (ii).∫(√((sin(x−α))/(sin(x+α)))) dx

$$\boldsymbol{\mathrm{Integrate}}\:\boldsymbol{\mathrm{following}}\:: \\ $$$$\:\:\left(\boldsymbol{\mathrm{i}}\right).\int\:\frac{\:\:\mathrm{dx}}{\mathrm{sin}\:\mathrm{x}\left(\:\mathrm{3}+\mathrm{2cos}\:\mathrm{x}\right)} \\ $$$$\:\:\left(\boldsymbol{\mathrm{ii}}\right).\int\sqrt{\frac{\mathrm{sin}\left(\mathrm{x}−\alpha\right)}{\mathrm{sin}\left(\mathrm{x}+\alpha\right)}}\:\:\mathrm{dx}\: \\ $$$$ \\ $$

Question Number 92799 Answers: 0 Comments: 2

Show that the function x→x^3 is of Riemann within the interval [−1,2] then calculate ∫_(−1) ^2 x^2 dx

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{x}\rightarrow\mathrm{x}^{\mathrm{3}} \:\mathrm{is} \\ $$$$\mathrm{of}\:\mathrm{Riemann}\:\mathrm{within}\:\mathrm{the}\:\mathrm{interval}\:\left[−\mathrm{1},\mathrm{2}\right] \\ $$$$\mathrm{then}\:\mathrm{calculate}\:\int_{−\mathrm{1}} ^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \mathrm{dx} \\ $$

Question Number 92795 Answers: 0 Comments: 0

Define Clairaut′s equation and solve y= px +(√(a^2 p^2 +b^2 ))

$$\boldsymbol{\mathrm{Define}}\:\boldsymbol{\mathrm{Clairaut}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{equation}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{solve}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{px}}\:+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} \boldsymbol{\mathrm{p}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} } \\ $$

Question Number 92790 Answers: 0 Comments: 0

∫_0 ^1 ((π/4)−tan^(−1) (x))(dx/(1−x^2 ))

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\pi}{\mathrm{4}}−{tan}^{−\mathrm{1}} \left({x}\right)\right)\frac{{dx}}{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$

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