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IntegrationQuestion and Answers: Page 23

Question Number 190552 Answers: 1 Comments: 1

∫_(1/2) ^2 ln(((ln(x+(1/x)))/(ln(x^2 −x+((17)/6)))))dx=?

$$\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{2}} {ln}\left(\frac{{ln}\left({x}+\frac{\mathrm{1}}{{x}}\right)}{{ln}\left({x}^{\mathrm{2}} −{x}+\frac{\mathrm{17}}{\mathrm{6}}\right)}\right){dx}=? \\ $$

Question Number 190533 Answers: 1 Comments: 0

Question Number 190487 Answers: 1 Comments: 0

Question Number 190419 Answers: 1 Comments: 0

∫_(−1) ^1 ∫_(−(√(1−y^2 ))) ^(√(1−y^2 )) ln (x^2 +y^2 +1)dx dy =?

$$\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\underset{−\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\overset{\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\int}}\:\mathrm{ln}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}\right){dx}\:{dy}\:=? \\ $$

Question Number 190385 Answers: 1 Comments: 0

Question Number 190371 Answers: 1 Comments: 0

Question Number 190360 Answers: 3 Comments: 0

if x+ (1/x) = ϕ ( Golden ratio) ⇒ x^( 2000) + (1/x^( 2000) )=?

$$ \\ $$$${if}\:\:\:{x}+\:\frac{\mathrm{1}}{{x}}\:=\:\varphi\:\left(\:\:{Golden}\:{ratio}\right) \\ $$$$\:\:\:\:\:\Rightarrow\:\:\:{x}^{\:\mathrm{2000}} +\:\frac{\mathrm{1}}{{x}^{\:\mathrm{2000}} }=? \\ $$$$ \\ $$$$ \\ $$

Question Number 190318 Answers: 2 Comments: 0

Question Number 190172 Answers: 1 Comments: 0

Integrate ∫_0 ^1 Sin^2 (2Πx)dx

$${Integrate}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{Sin}^{\mathrm{2}} \left(\mathrm{2}\Pi{x}\right){dx} \\ $$

Question Number 190168 Answers: 0 Comments: 2

1)∫^∞ _0 ((sin x)/(x^p +sin x))dx ,p>0 2)∫^∞ _π ((xcos x)/(x^p +x^q ))dx,p>0and q>0 3)∫^∞ _0 ((sin x^p )/( x^q ))dx, p>0,q>0 4)∫^2 _0 (dx/(∣ln x∣^p )) ,p>0 5)∫^1 _0 ((cos(1/(1−x)))/( ((1−x^2 ))^(1/n) ))dx 6)∫^∞ _0 (dx/(x^p ((sin^2 x))^(1/3) ))

$$\left.\mathrm{1}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{sin}\:{x}}{{x}^{{p}} +{sin}\:{x}}{dx}\:,{p}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\underset{\pi} {\int}^{\infty} \frac{{xcos}\:{x}}{{x}^{{p}} +{x}^{{q}} }{dx},{p}>\mathrm{0}{and}\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{sin}\:{x}^{{p}} }{\:{x}^{{q}} }{dx},\:{p}>\mathrm{0},{q}>\mathrm{0} \\ $$$$\left.\mathrm{4}\right)\underset{\mathrm{0}} {\int}^{\mathrm{2}} \frac{{dx}}{\mid{ln}\:{x}\mid^{{p}} }\:,{p}>\mathrm{0} \\ $$$$\left.\mathrm{5}\right)\underset{\mathrm{0}} {\int}^{\mathrm{1}} \frac{{cos}\frac{\mathrm{1}}{\mathrm{1}−{x}}}{\:\sqrt[{{n}}]{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$$$\left.\mathrm{6}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{dx}}{{x}^{{p}} \sqrt[{\mathrm{3}}]{{sin}^{\mathrm{2}} {x}}} \\ $$

Question Number 190009 Answers: 0 Comments: 2

Question Number 189949 Answers: 1 Comments: 0

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

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