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IntegrationQuestion and Answers: Page 201

Question Number 82439 Answers: 0 Comments: 1

calculate I_n =∫_0 ^1 x^n (√(1+x+x^2 ))dx

$${calculate}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 82442 Answers: 0 Comments: 1

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

$$\left.\mathrm{1}\right){find}\:\int\:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 82435 Answers: 0 Comments: 1

calculate ∫_4 ^(+∞) (x^3 /((2x+1)^3 (x−3)^5 ))dx

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

Question Number 82434 Answers: 0 Comments: 0

1)decompose inside C(x)and R(x) the fraction F(x)=((2x+1)/((x^2 +1)^3 (x−1)^2 )) 2) find the value of ∫_3 ^(+∞) F(x)dx

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left({x}\right){and}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{3}} ^{+\infty} {F}\left({x}\right){dx} \\ $$

Question Number 82433 Answers: 0 Comments: 1

1)decompose inside C(x)and R(x) F=(1/((x^2 +x+1)^2 )) 2)calculate ∫_0 ^∞ (dx/((x^2 +x+1)^2 ))

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left({x}\right){and}\:{R}\left({x}\right)\:{F}=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 82402 Answers: 0 Comments: 0

Question Number 82391 Answers: 0 Comments: 1

∫ sin x cos (sin x) dx ?

$$\int\:\mathrm{sin}\:{x}\:\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)\:{dx}\:? \\ $$

Question Number 82330 Answers: 0 Comments: 4

Question Number 82286 Answers: 1 Comments: 3

1) find a and b wich verify ∫_0 ^π (at^2 +bt)cos(nx) =(1/n^2 ) 2) find the value of Σ_(n=1) ^∞ (1/n^2 )

$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{wich}\:{verify}\:\:\int_{\mathrm{0}} ^{\pi} \left({at}^{\mathrm{2}} \:+{bt}\right){cos}\left({nx}\right)\:=\frac{\mathrm{1}}{{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} } \\ $$

Question Number 82283 Answers: 1 Comments: 2

∫x^3 (√(x^3 +1)) dx

$$\int{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx} \\ $$

Question Number 82244 Answers: 1 Comments: 2

find the function of f when this function continue at interval [−∞,0] ∫_(−x^2 ) ^0 f(t) dt=(d/dx)[x(1−sin(πx)]

$${find}\:{the}\:{function}\:{of}\:{f}\:{when}\:{this}\:\: \\ $$$${function}\:{continue}\:{at}\:{interval}\:\left[−\infty,\mathrm{0}\right] \\ $$$$\int_{−{x}^{\mathrm{2}} } ^{\mathrm{0}} {f}\left({t}\right)\:{dt}=\frac{{d}}{{dx}}\left[{x}\left(\mathrm{1}−{sin}\left(\pi{x}\right)\right]\right. \\ $$

Question Number 82232 Answers: 1 Comments: 1

Question Number 82223 Answers: 0 Comments: 1

given f(x) = f(x+(π/6)) , ∀x∈ R if ∫_0 ^(π/6) f(x) dx = T then ∫_π ^((7π)/3) f(x+π) dx = ?

$${given}\:{f}\left({x}\right)\:=\:{f}\left({x}+\frac{\pi}{\mathrm{6}}\right)\:,\:\forall{x}\in\:\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{6}}} {\int}}\:{f}\left({x}\right)\:{dx}\:=\:{T}\: \\ $$$${then}\:\underset{\pi} {\overset{\frac{\mathrm{7}\pi}{\mathrm{3}}} {\int}}\:{f}\left({x}+\pi\right)\:{dx}\:=\:? \\ $$

Question Number 82185 Answers: 1 Comments: 2

∫ (dx/(sec x + csc x)) = ?

$$\int\:\frac{{dx}}{\mathrm{sec}\:{x}\:+\:{csc}\:{x}}\:=\:?\: \\ $$

Question Number 82174 Answers: 1 Comments: 1

∫_0 ^π x ln(sin x) dx = ?

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:{x}\:{ln}\left(\mathrm{sin}\:{x}\right)\:{dx}\:=\:?\: \\ $$

Question Number 82139 Answers: 1 Comments: 0

∫ ((√(x^4 +x^(−4) +2))/x^3 ) dx

$$\int\:\:\frac{\sqrt{{x}^{\mathrm{4}} +{x}^{−\mathrm{4}} +\mathrm{2}}}{{x}^{\mathrm{3}} }\:{dx}\: \\ $$

Question Number 82022 Answers: 0 Comments: 0

Question Number 82020 Answers: 0 Comments: 2

Question Number 81996 Answers: 0 Comments: 0

calculate I_n =∫∫_([(1/n),n[) e^(−x^2 −3y^2 ) dxdy and find lim_(n→+∞) I_n conclude that ∫_0 ^∞ e^(−x^2 ) dx=((√π)/2)

$${calculate}\:{I}_{{n}} =\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[\right.\right.} \:\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } {dxdy} \\ $$$${and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:{I}_{{n}} \\ $$$${conclude}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$

Question Number 81994 Answers: 0 Comments: 0

calculate ∫∫_W (x+y)e^(x−y) dxdy with W is the triangle limited by o,A(1,0)and B(0,1)

$${calculate}\:\int\int_{{W}} \left({x}+{y}\right){e}^{{x}−{y}} {dxdy} \\ $$$${with}\:{W}\:{is}\:{the}\:{triangle}\:{limited}\:{by} \\ $$$${o},{A}\left(\mathrm{1},\mathrm{0}\right){and}\:{B}\left(\mathrm{0},\mathrm{1}\right) \\ $$

Question Number 81993 Answers: 0 Comments: 0

calculate ∫∫_D ln(1+x+y)dxdy with D is the triangle limited by points 0,A(1,0) and B(0,1)

$${calculate}\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy} \\ $$$${with}\:{D}\:{is}\:{the}\:{triangle}\:{limited}\:{by} \\ $$$${points}\:\mathrm{0},{A}\left(\mathrm{1},\mathrm{0}\right)\:{and}\:{B}\left(\mathrm{0},\mathrm{1}\right) \\ $$

Question Number 81921 Answers: 1 Comments: 0

Question Number 81889 Answers: 1 Comments: 0

Question Number 81888 Answers: 1 Comments: 1

Question Number 81801 Answers: 1 Comments: 1

Question Number 81739 Answers: 2 Comments: 1

∫(dx/(cos^3 x−sin^3 x))

$$\int\frac{{dx}}{{cos}^{\mathrm{3}} {x}−{sin}^{\mathrm{3}} {x}} \\ $$

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