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IntegrationQuestion and Answers: Page 280

Question Number 39840    Answers: 0   Comments: 1

calculate lim_(x→0) ∫_(x+1) ^(x^2 +1) ln(1+t) e^(−t) dt

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\int_{{x}+\mathrm{1}} ^{{x}^{\mathrm{2}} \:+\mathrm{1}} \:\:{ln}\left(\mathrm{1}+{t}\right)\:{e}^{−{t}} {dt}\: \\ $$

Question Number 39838    Answers: 0   Comments: 1

find lim_(ξ→0) ∫_0 ^1 (dx/((√(1+ξx^2 ))−(√(1−ξx^2 ))))

$${find}\:{lim}_{\xi\rightarrow\mathrm{0}} \:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+\xi{x}^{\mathrm{2}} }−\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}\: \\ $$

Question Number 39836    Answers: 0   Comments: 0

calculate ∫_0 ^(+∞) ((ln(1+ix^2 ))/(2+x^2 ))dx

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

Question Number 39834    Answers: 1   Comments: 0

calculate ∫_0 ^(π/6) ∣ cos(2x)−cos(3x)∣dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\mid\:{cos}\left(\mathrm{2}{x}\right)−{cos}\left(\mathrm{3}{x}\right)\mid{dx} \\ $$

Question Number 39833    Answers: 2   Comments: 1

find ∫ ((ln(x+(√(x^2 −1))))/(√(x^2 −1))) dx 2) calculate ∫_2 ^5 ((ln(x+(√(x^2 −1)))/(√(x^2 −1)))dx

$${find}\:\:\int\:\:\:\:\frac{{ln}\left({x}+\sqrt{{x}^{\mathrm{2}} \:−\mathrm{1}}\right)}{\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{2}} ^{\mathrm{5}} \:\:\:\frac{{ln}\left({x}+\sqrt{{x}^{\mathrm{2}} \:−\mathrm{1}}\right.}{\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{dx} \\ $$

Question Number 40139    Answers: 0   Comments: 3

let I_n = ∫_0 ^1 x^n (√(1−x)) dx 1) calculate I_0 and I_1 2) prove that ∀n∈ N^★ (3+2n) I_n =2n I_(n−1) 3) find I_n interms of n

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{x}^{{n}} \sqrt{\mathrm{1}−{x}}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{\mathrm{0}} \:\:{and}\:{I}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\forall{n}\in\:{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right)\:{I}_{{n}} =\mathrm{2}{n}\:{I}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}_{{n}} \:\:{interms}\:{of}\:{n} \\ $$

Question Number 39787    Answers: 0   Comments: 2

calculste I_λ = ∫_(−∞) ^(+∞) ((cos(λx^n ))/(1+x^2 )) dx with λ from R and n integr natural 2) find the vslue of ∫_(−∞) ^(+∞) ((cos(3 x^9 ))/(1+x^2 )) dx .

$${calculste}\:\:{I}_{\lambda} \:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left(\lambda{x}^{{n}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with} \\ $$$$\lambda\:{from}\:{R}\:{and}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{vslue}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left(\mathrm{3}\:{x}^{\mathrm{9}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 39712    Answers: 1   Comments: 3

calculate ∫_(−∞) ^(+∞) ((cos(x^n ) +sin(x^n ))/((x^2 +9)^n )) dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left({x}^{{n}} \right)\:+{sin}\left({x}^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{9}\right)^{{n}} }\:{dx} \\ $$

Question Number 39711    Answers: 0   Comments: 1

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

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{{n}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:{dx}\:{with}\:{n}\:{natral}\:{integr} \\ $$

Question Number 39706    Answers: 0   Comments: 0

Question Number 39660    Answers: 0   Comments: 2

let S_n = ∫_0 ^n ((x(−1)^([x]) )/((x+1 −[x])^3 ))dx 1) calculate S_n 2) find lim_(n→+∞) S_n

$${let}\:\:{S}_{{n}} \:\:=\:\int_{\mathrm{0}} ^{{n}} \:\:\:\:\:\frac{{x}\left(−\mathrm{1}\right)^{\left[{x}\right]} }{\left({x}+\mathrm{1}\:−\left[{x}\right]\right)^{\mathrm{3}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{S}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} \\ $$

Question Number 39633    Answers: 0   Comments: 3

find the value of f(x) = ∫_0 ^π ln(x^2 −2x cosθ +1)dθ with x fromR.

$${find}\:{the}\:{value}\:{of}\: \\ $$$${f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} \:−\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:{x}\:{fromR}. \\ $$

Question Number 39443    Answers: 1   Comments: 3

lim_(n→∞) [ (1/(n^2 +1))+ (2/(n^2 +2))+ (3/(n^2 +3))+ ....+(1/(n+1))] = ?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{2}}+\:\frac{\mathrm{3}}{\mathrm{n}^{\mathrm{2}} +\mathrm{3}}+\:....+\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right]\:=\:? \\ $$

Question Number 39441    Answers: 0   Comments: 2

∫_(1/4) ^( 4) (1/x) sin (x−(1/x))dx = ?

$$\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\:\mathrm{4}} \:\frac{\mathrm{1}}{{x}}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:=\:? \\ $$

Question Number 39440    Answers: 1   Comments: 0

f(x)= ∫_0 ^( x_ ) e^(t ) (((1+sin t)/(1+cos t))) dt. Then f((π/3))×f(((2π)/3)) = ?

$$\mathrm{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\:{x}_{} } \:{e}^{{t}\:} \left(\frac{\mathrm{1}+\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}\right)\:{dt}. \\ $$$${T}\mathrm{hen}\:\:\mathrm{f}\left(\frac{\pi}{\mathrm{3}}\right)×\mathrm{f}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\:=\:? \\ $$

Question Number 39477    Answers: 1   Comments: 3

∫2^x 3^(2x) dx=?

$$\int\mathrm{2}^{\mathrm{x}} \mathrm{3}^{\mathrm{2x}} \mathrm{dx}=? \\ $$

Question Number 39431    Answers: 1   Comments: 1

∫_0 ^(2π) e^(x/2) sin ((x/2)+(π/4))dx = ?

$$\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\mathrm{e}^{\frac{{x}}{\mathrm{2}}} \mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\mathrm{d}{x}\:=\:? \\ $$

Question Number 39483    Answers: 0   Comments: 3

find f(t)= ∫_0 ^1 ((ln(1+xt))/(1+x^2 )) dx .

$${find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 39389    Answers: 0   Comments: 2

calculate F(x) = ∫_0 ^∞ (dt/(1+(1+x(1+t^2 ))^2 ))

$${calculate}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{dt}}{\mathrm{1}+\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)^{\mathrm{2}} } \\ $$

Question Number 39386    Answers: 1   Comments: 1

find the value of ∫_0 ^1 ((ln(1+x))/(1+x^2 ))dx

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

Question Number 39384    Answers: 2   Comments: 0

The values of a for which y= ax^2 +ax+(1/(24)) and x = ay^2 +ay+(1/(24)) touch each other are 1) (2/3) 2) (3/2) 3) ((13+(√(601)))/(12)) 4) ((13−(√(601)))/(12)).

$$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{for}\:\mathrm{which}\:\mathrm{y}=\:\mathrm{a}{x}^{\mathrm{2}} +{ax}+\frac{\mathrm{1}}{\mathrm{24}} \\ $$$${and}\:{x}\:=\:{ay}^{\mathrm{2}} +{ay}+\frac{\mathrm{1}}{\mathrm{24}}\:{touch}\:{each}\:{other} \\ $$$${are} \\ $$$$\left.\mathrm{1}\left.\right)\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\right)\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\left.\right)\:\frac{\mathrm{13}+\sqrt{\mathrm{601}}}{\mathrm{12}}\:\:\:\:\:\:\:\mathrm{4}\right)\:\frac{\mathrm{13}−\sqrt{\mathrm{601}}}{\mathrm{12}}. \\ $$

Question Number 39383    Answers: 1   Comments: 1

calculate ∫_0 ^(π/3) ((sinxdx)/(cosx(2+ln(cosx))) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\:\:\frac{{sinxdx}}{{cosx}\left(\mathrm{2}+{ln}\left({cosx}\right)\right.}\:. \\ $$

Question Number 39382    Answers: 1   Comments: 0

Question Number 39381    Answers: 1   Comments: 0

Question Number 39379    Answers: 1   Comments: 6

Question Number 39375    Answers: 0   Comments: 1

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