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IntegrationQuestion and Answers: Page 222

Question Number 67466 Answers: 0 Comments: 0

let consider for all n≥1 the real (t)_n =t(t+1).....(t+n−1) Find L_n = ∫_0 ^∞ (((t)_1 )/((t)_(n+1) )) dt

$$ \\ $$$$ \\ $$$${let}\:{consider}\:\:\:{for}\:{all}\:{n}\geqslant\mathrm{1}\:{the}\:{real}\:\left({t}\right)_{{n}} \:={t}\left({t}+\mathrm{1}\right).....\left({t}+{n}−\mathrm{1}\right) \\ $$$${Find}\:\:\:{L}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left({t}\right)_{\mathrm{1}} }{\left({t}\right)_{{n}+\mathrm{1}} }\:{dt} \\ $$

Question Number 67463 Answers: 1 Comments: 3

Find Find K=∫_0 ^(π/2) (√(tanθ)) dθ

$${Find} \\ $$$${Find}\:\:\:{K}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{tan}\theta}\:{d}\theta\: \\ $$

Question Number 67462 Answers: 0 Comments: 2

Calculate when a,b are positive reals f(a,b)= ∫_0 ^1 ((t^a −t^b )/(lnt)) dt

$$ \\ $$$$\:{Calculate}\:{when}\:{a},{b}\:{are}\:{positive}\:{reals}\:\:\:{f}\left({a},{b}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{{a}} −{t}^{{b}} }{{lnt}}\:{dt}\: \\ $$

Question Number 68043 Answers: 1 Comments: 1

∫_(π/2) ^π e^(cosx) (√(1−e^(cosx) )) sinx dx

$$\int_{\pi/\mathrm{2}} ^{\pi} {e}^{{cosx}} \sqrt{\mathrm{1}−{e}^{{cosx}} }\:{sinx}\:{dx} \\ $$

Question Number 67392 Answers: 0 Comments: 0

∫(6x

$$\int\left(\mathrm{6}{x}\right. \\ $$

Question Number 67385 Answers: 0 Comments: 0

find ∫ x((√((1−x^2 )/(1+x^2 ))))dx

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

Question Number 67374 Answers: 0 Comments: 3

find ∫ (1+(1/x^2 ))arctan(1−(1/x))dx

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

Question Number 67373 Answers: 1 Comments: 4

simplify S_n (x) =Σ_(k=0) ^n C_n ^k cos^4 (πkx) 2) calculate I_n =∫_0 ^(1/3) S_n (x)dx

$${simplify}\:\:\:{S}_{{n}} \left({x}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{\mathrm{4}} \left(\pi{kx}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{3}}} \:{S}_{{n}} \left({x}\right){dx} \\ $$

Question Number 67359 Answers: 1 Comments: 1

∫siny/y dy

$$\int{siny}/{y}\:\:{dy} \\ $$

Question Number 67342 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^∞ ((sin(2x^2 ))/((x^2 −x +3)^3 ))dx

$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} −{x}\:+\mathrm{3}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 67310 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/(x^4 +x^2 +1))

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 67246 Answers: 2 Comments: 4

Integrate: 1) ∫_3 ^( ∞) ((1/x dx)/(ln(x)(√(ln^2 x−1)))) 2) ∫_1 ^∞ ((e^x dx)/(1+e^(2x) )) 3) ∫_1 ^∞ ((2^x dx)/(x+1)) 4) ∫_2 ^∞ ((√x)/(ln(x)))dx

$${Integrate}: \\ $$$$\left.\mathrm{1}\right)\:\underset{\mathrm{3}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}/{x}\:{dx}}{{ln}\left({x}\right)\sqrt{{ln}^{\mathrm{2}} {x}−\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{e}^{{x}} {dx}}{\mathrm{1}+{e}^{\mathrm{2}{x}} } \\ $$$$\left.\mathrm{3}\right)\:\underset{\mathrm{1}} {\overset{\infty} {\int}}\:\frac{\mathrm{2}^{{x}} {dx}}{{x}+\mathrm{1}} \\ $$$$\left.\mathrm{4}\right)\:\underset{\mathrm{2}} {\overset{\infty} {\int}}\:\frac{\sqrt{{x}}}{{ln}\left({x}\right)}{dx} \\ $$

Question Number 67235 Answers: 0 Comments: 1

find ∫_(−(π/3)) ^(π/3) x^2 {cosx−sinx}^3 dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{3}}} \:{x}^{\mathrm{2}} \left\{{cosx}−{sinx}\right\}^{\mathrm{3}} {dx} \\ $$

Question Number 67233 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((xdx)/(√(1+x^4 )))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$

Question Number 67231 Answers: 2 Comments: 0

find ∫x/x^5 −1) dx

$$\left.{find}\:\int{x}/{x}^{\mathrm{5}} −\mathrm{1}\right)\:{dx} \\ $$

Question Number 67197 Answers: 1 Comments: 0

∫_0 ^2 x^5 (1−(x/2))^4 dx

$$\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{5}} \left(\mathrm{1}−\frac{{x}}{\mathrm{2}}\right)^{\mathrm{4}} {dx} \\ $$

Question Number 67153 Answers: 2 Comments: 0

find ∫(v^3 −2)/(v^4 +v )dv

$${find}\:\int\left({v}^{\mathrm{3}} −\mathrm{2}\right)/\left({v}^{\mathrm{4}} +{v}\:\:\right){dv} \\ $$

Question Number 67138 Answers: 0 Comments: 1

find the area abounded y=(√x) and y=x−2?

$${find}\:{the}\:{area}\:{abounded}\:{y}=\sqrt{{x}} \\ $$$${and}\:{y}={x}−\mathrm{2}? \\ $$

Question Number 67106 Answers: 0 Comments: 1

find the area abounded y=(√x) afind y=x−2?

$${find}\:{the}\:{area}\:{abounded}\:{y}=\sqrt{{x}} \\ $$$${afind}\:{y}={x}−\mathrm{2}? \\ $$

Question Number 67105 Answers: 0 Comments: 0

find the area abounded y=(√x) afind y=x−2?

$${find}\:{the}\:{area}\:{abounded}\:{y}=\sqrt{{x}} \\ $$$${afind}\:{y}={x}−\mathrm{2}? \\ $$

Question Number 67070 Answers: 0 Comments: 3

Question Number 67069 Answers: 0 Comments: 1

Question Number 67059 Answers: 0 Comments: 1

find the area abounded y=(√(x−2)) and y=x−2 ?

$${find}\:{the}\:{area}\:{abounded}\:{y}=\sqrt{{x}−\mathrm{2}} \\ $$$${and}\:{y}={x}−\mathrm{2}\:? \\ $$

Question Number 67058 Answers: 0 Comments: 0

find the area abounded y=(√(x−2)) and y=x−2 ?

$${find}\:{the}\:{area}\:{abounded}\:{y}=\sqrt{{x}−\mathrm{2}} \\ $$$${and}\:{y}={x}−\mathrm{2}\:? \\ $$

Question Number 67038 Answers: 1 Comments: 1

calculate ∫_(−1) ^1 (x^(2n) /(1+2^(sinx) ))dx with n integr.

$${calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\frac{{x}^{\mathrm{2}{n}} }{\mathrm{1}+\mathrm{2}^{{sinx}} }{dx}\:\:\:{with}\:{n}\:{integr}. \\ $$

Question Number 67035 Answers: 1 Comments: 2

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