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IntegrationQuestion and Answers: Page 20

Question Number 189890 Answers: 0 Comments: 0

Question Number 189825 Answers: 2 Comments: 0

∫^b _a (√((x−a)(b−x)))=¿

$$\underset{{a}} {\int}^{{b}} \sqrt{\left({x}−{a}\right)\left({b}−{x}\right)}=¿ \\ $$

Question Number 189756 Answers: 0 Comments: 0

Question Number 189752 Answers: 1 Comments: 0

∫^∞ _0 x^2 .e^(−x^2 ) dx=¿

$$\underset{\mathrm{0}} {\int}^{\infty} {x}^{\mathrm{2}} .{e}^{−{x}^{\mathrm{2}} } {dx}=¿ \\ $$

Question Number 189639 Answers: 1 Comments: 1

(((20)),(( 0)) ) (((10)),(( 1)) ) + (((20)),(( 1)) ) (((10)),(( 2)) ) +...+ ((( 20)),(( 9)) ) ((( 10)),(( 10)) ) =?

$$ \\ $$$$\:\:\begin{pmatrix}{\mathrm{20}}\\{\:\mathrm{0}}\end{pmatrix}\:\begin{pmatrix}{\mathrm{10}}\\{\:\mathrm{1}}\end{pmatrix}\:+\:\begin{pmatrix}{\mathrm{20}}\\{\:\mathrm{1}}\end{pmatrix}\:\begin{pmatrix}{\mathrm{10}}\\{\:\:\mathrm{2}}\end{pmatrix}\:+...+\:\begin{pmatrix}{\:\:\:\mathrm{20}}\\{\:\:\mathrm{9}}\end{pmatrix}\:\begin{pmatrix}{\:\:\mathrm{10}}\\{\:\mathrm{10}}\end{pmatrix}\:=? \\ $$$$ \\ $$

Question Number 189549 Answers: 1 Comments: 0

∫_0 ^( 1) ∫_0 ^( 1) ((dxdy)/((1+xy )^( 4) ))=?

$$ \\ $$$$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{dxdy}}{\left(\mathrm{1}+{xy}\:\right)^{\:\mathrm{4}} }=? \\ $$

Question Number 189496 Answers: 2 Comments: 0

∫ xe^(x^2 /2) dx

$$\int\:\mathrm{xe}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{dx} \\ $$

Question Number 189489 Answers: 1 Comments: 1

Ω= ∫_0 ^( ∞) e^( −x) cos(x)ln(x)dx=? −−− f (a )= ∫_0 ^( ∞) e^( −x) cos(x)x^( a) dx = Re ∫_0 ^( ∞) e^( −x) .e^( −ix) .x^( a) dx = Re ∫_0 ^( ∞) e^( −x (1+i)) .x^( a) dx = Re(L { x^( a) }∣_( s= i+1) ) = Re( ((Γ (1+a))/s^( a+1) ) ∣_( 1+i) = ((Γ (1+a))/((1+i)^( a+1) )) ) Re (Γ(1+a).2^( ((1+a)/2)) . e^( −((iπ)/4) (1+a)) ) Ω= f ′(0)=.......

$$ \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} {e}^{\:−{x}} {cos}\left({x}\right){ln}\left({x}\right){dx}=? \\ $$$$\:\:\:\:\:−−− \\ $$$$\:\:\:\:\:\:{f}\:\left({a}\:\right)=\:\int_{\mathrm{0}} ^{\:\infty} {e}^{\:−{x}} {cos}\left({x}\right){x}^{\:{a}} \:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{Re}\:\int_{\mathrm{0}} ^{\:\infty} {e}^{\:−{x}} .{e}^{\:−{ix}} .{x}^{\:{a}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:{Re}\:\int_{\mathrm{0}} ^{\:\infty} \:{e}^{\:−{x}\:\left(\mathrm{1}+{i}\right)} .{x}^{\:{a}} \:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:{Re}\left(\mathscr{L}\:\:\left\{\:{x}^{\:{a}} \:\right\}\mid_{\:{s}=\:{i}+\mathrm{1}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:=\:{Re}\left(\:\frac{\Gamma\:\left(\mathrm{1}+{a}\right)}{{s}^{\:{a}+\mathrm{1}} }\:\mid_{\:\mathrm{1}+{i}} =\:\frac{\Gamma\:\left(\mathrm{1}+{a}\right)}{\left(\mathrm{1}+{i}\right)^{\:{a}+\mathrm{1}} }\:\right) \\ $$$$\:\:\:\:\:\:\:\:{Re}\:\left(\Gamma\left(\mathrm{1}+{a}\right).\mathrm{2}^{\:\frac{\mathrm{1}+{a}}{\mathrm{2}}} .\:{e}^{\:−\frac{{i}\pi}{\mathrm{4}}\:\left(\mathrm{1}+{a}\right)} \right) \\ $$$$\:\:\:\:\Omega=\:{f}\:'\left(\mathrm{0}\right)=....... \\ $$$$\:\: \\ $$

Question Number 189418 Answers: 1 Comments: 0

Know: f(x)=3x+2+∫^1 _0 xf(x)dx Eluavte: ∫^2 _0 f(x)dx=¿

$${Know}:\:{f}\left({x}\right)=\mathrm{3}{x}+\mathrm{2}+\underset{\mathrm{0}} {\int}^{\mathrm{1}} {xf}\left({x}\right){dx} \\ $$$${Eluavte}:\:\underset{\mathrm{0}} {\int}^{\mathrm{2}} {f}\left({x}\right){dx}=¿ \\ $$

Question Number 189390 Answers: 0 Comments: 0

Question Number 189345 Answers: 1 Comments: 0

solve ∫t^(−6) (t^2 +3)^2 dt

$${solve} \\ $$$$\int{t}^{−\mathrm{6}} \left({t}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} {dt} \\ $$

Question Number 189325 Answers: 2 Comments: 0

prove Ω= ∫_0 ^(π/2) (( cos(x)+cos(5x))/(1+ 2sin(x))) =^( ?) (3/2)

$$ \\ $$$$\:\:\:\:\:{prove} \\ $$$$ \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\:{cos}\left({x}\right)+{cos}\left(\mathrm{5}{x}\right)}{\mathrm{1}+\:\mathrm{2}{sin}\left({x}\right)}\:\overset{\:?} {=}\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 189323 Answers: 1 Comments: 0

Question Number 189293 Answers: 1 Comments: 0

Question Number 189266 Answers: 1 Comments: 0

∫_0 ^(π/2) (((tan x))^(1/3) /(1+sin 2x)) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{tan}\:\mathrm{x}}}{\mathrm{1}+\mathrm{sin}\:\mathrm{2x}}\:\mathrm{dx}\:=? \\ $$

Question Number 189250 Answers: 0 Comments: 1

Question Number 189189 Answers: 1 Comments: 0

Question Number 189144 Answers: 0 Comments: 1

∫_0 ^( 1) ∫_0 ^( 1) ∫_0 ^( 1) ((√(x + y + z))/( (√x) + (√y) + (√z) )) dxdydz

$$\: \\ $$$$\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\sqrt{{x}\:+\:{y}\:+\:{z}}}{\:\sqrt{{x}}\:+\:\sqrt{{y}}\:+\:\sqrt{{z}}\:}\:{dxdydz} \\ $$$$\: \\ $$$$\: \\ $$

Question Number 189066 Answers: 2 Comments: 0

Given f(x)+∫_0 ^1 (x+y)^2 f(y) dy=2x^2 −3x+1 find f(x).

$$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{2}} \:\mathrm{f}\left(\mathrm{y}\right)\:\mathrm{dy}=\mathrm{2x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{1} \\ $$$$\:\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$

Question Number 189057 Answers: 0 Comments: 6

Help! Evaluate the following integral usings Green theorem: ∮4xydx + x^2 dy Where C is the square of vertices (0,0), (0,2), (2,0) and (2,2).

$$\: \\ $$$$\:\mathrm{Help}! \\ $$$$\: \\ $$$$\:\mathrm{Evaluate}\:\:\mathrm{the}\:\:\mathrm{following}\:\:\mathrm{integral}\:\:\mathrm{usings}\:\:\mathrm{Green}\:\mathrm{theorem}: \\ $$$$\: \\ $$$$\:\oint\mathrm{4xy}{d}\mathrm{x}\:\:+\:\:\mathrm{x}^{\mathrm{2}} {d}\mathrm{y} \\ $$$$\: \\ $$$$\:\mathrm{Where}\:\:{C}\:\:\mathrm{is}\:\:\mathrm{the}\:\:\mathrm{square}\:\:\mathrm{of}\:\:\mathrm{vertices}\:\:\left(\mathrm{0},\mathrm{0}\right),\:\left(\mathrm{0},\mathrm{2}\right),\:\left(\mathrm{2},\mathrm{0}\right)\:\:\mathrm{and}\:\:\left(\mathrm{2},\mathrm{2}\right). \\ $$$$\: \\ $$

Question Number 188982 Answers: 0 Comments: 2

Question Number 188889 Answers: 1 Comments: 0

If, y= (( Arcsin((√x) ))/( (√( x (1−x ))))) ⇒ y′ .p(x) + y .q(x)= 1 find , ∫_0 ^( 1) p(x).q(x)dx=? p , q are two pllynomils...

$$ \\ $$$$\:\:{If},\:{y}=\:\frac{\:{Arcsin}\left(\sqrt{{x}}\:\right)}{\:\sqrt{\:{x}\:\left(\mathrm{1}−{x}\:\right)}}\:\:\Rightarrow \\ $$$$\:\:\:{y}'\:.{p}\left({x}\right)\:+\:{y}\:.{q}\left({x}\right)=\:\mathrm{1} \\ $$$$ \\ $$$$\:\:\:{find}\:,\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {p}\left({x}\right).{q}\left({x}\right){dx}=? \\ $$$$\:\:\:\:{p}\:,\:{q}\:\:{are}\:{two}\:{pllynomils}... \\ $$$$ \\ $$

Question Number 188861 Answers: 2 Comments: 0

Question Number 188552 Answers: 0 Comments: 1

Question Number 188511 Answers: 0 Comments: 0

evaluate ∫_0 ^π (dx/(a+bcosx )) , a > 0 and deduce that ∫_0 ^π (dx/((a+bcos x)^2 )) = ((πa)/((a^2 −b^2 )^(3/2) )) ; a^2 >b^2 and ∫_0 ^π ((cos x dx)/((a+bcos x)^2 )) = ((−πb)/((a^2 −b^2 )^(3/2) )) ; a^2 >b^2

$$\:\:\:{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{{a}+{b}\mathrm{cos}{x}\:}\:\:\:\:\:\:,\:\:\:{a}\:>\:\mathrm{0} \\ $$$$\:\:\:{and}\:{deduce}\:{that} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\:\:\frac{\pi{a}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$$${and}\:\:\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}\:{x}\:{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\frac{−\pi{b}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$

Question Number 188470 Answers: 0 Comments: 0

∫_0 ^(π/4) arctan((√((1−tan^2 x)/2)))dx = ?

$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}{arctan}\left(\sqrt{\frac{\mathrm{1}−{tan}^{\mathrm{2}} {x}}{\mathrm{2}}}\right){dx}\:=\:? \\ $$

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