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IntegrationQuestion and Answers: Page 248

Question Number 50420 Answers: 1 Comments: 4

find ∫_0 ^(π/6) cosx ln(cosx)dx

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\:{cosx}\:{ln}\left({cosx}\right){dx} \\ $$

Question Number 50418 Answers: 0 Comments: 1

calculate ∫_0 ^(lln(3)) ((sh^2 (x)dx)/(ch^3 (x)))

$${calculate}\:\int_{\mathrm{0}} ^{{lln}\left(\mathrm{3}\right)} \:\:\frac{{sh}^{\mathrm{2}} \left({x}\right){dx}}{{ch}^{\mathrm{3}} \left({x}\right)} \\ $$

Question Number 50417 Answers: 0 Comments: 1

find ∫_0 ^1 arctan(√(1−(x^2 /2)))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}{dx} \\ $$

Question Number 50416 Answers: 1 Comments: 0

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

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)}{dx} \\ $$

Question Number 50415 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) (dt/(1+cosθ cost))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dt}}{\mathrm{1}+{cos}\theta\:{cost}} \\ $$

Question Number 50414 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) ((x sinx cosx)/(tan^2 x +cotan^2 x))dx ctanx =(1/(tanx))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}\:{sinx}\:{cosx}}{{tan}^{\mathrm{2}} {x}\:+{cotan}^{\mathrm{2}} {x}}{dx} \\ $$$${ctanx}\:=\frac{\mathrm{1}}{{tanx}} \\ $$

Question Number 50413 Answers: 0 Comments: 1

let f ∈C^0 (R,R) / ∀ x∈R f(a+b−x)=f(x) 1) find ∫_a ^b xf(x)dx interms of ∫_a ^b f(x)dx 2) calculate ∫_0 ^π ((xdx)/(1+sinx))

$${let}\:{f}\:\in{C}^{\mathrm{0}} \left({R},{R}\right)\:/\:\forall\:{x}\in{R}\:\:{f}\left({a}+{b}−{x}\right)={f}\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:\int_{{a}} ^{{b}} {xf}\left({x}\right){dx}\:{interms}\:{of}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{xdx}}{\mathrm{1}+{sinx}} \\ $$

Question Number 50412 Answers: 0 Comments: 1

1) calculate U_n =∫_0 ^π (dx/(1+cos^2 (nx))) with n from N 2) f continue from [0,π] to R find lim_(n→+∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx

$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:{with}\:{n}\:{from}\:{N} \\ $$$$\left.\mathrm{2}\right)\:{f}\:{continue}\:{from}\:\left[\mathrm{0},\pi\right]\:{to}\:{R}\:\:{find} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx} \\ $$

Question Number 50410 Answers: 0 Comments: 0

determine all functions f ∈C^0 (R,R) / ∫_0 ^x f(x)dx =(2/3)xf(x) .

$${determine}\:{all}\:{functions}\:{f}\:\in{C}^{\mathrm{0}} \left({R},{R}\right)\:/ \\ $$$$\int_{\mathrm{0}} ^{{x}} {f}\left({x}\right){dx}\:=\frac{\mathrm{2}}{\mathrm{3}}{xf}\left({x}\right)\:. \\ $$

Question Number 50407 Answers: 0 Comments: 0

determine f ∈C^0 ([0,1],R) verifying ∫_0 ^1 f(x)dx =(1/3) +∫_0 ^1 (f(x^2 ))^2 dx

$${determine}\:{f}\:\in{C}^{\mathrm{0}} \left(\left[\mathrm{0},\mathrm{1}\right],{R}\right)\:{verifying} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\frac{\mathrm{1}}{\mathrm{3}}\:+\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left({f}\left({x}^{\mathrm{2}} \right)\right)^{\mathrm{2}} {dx} \\ $$

Question Number 50406 Answers: 0 Comments: 2

1) decompose at simple elements U_n = ((n x^(n−1) )/(x^n −1)) 2) calculste ∫_0 ^(2π) (dt/(x−e^(it) ))

$$\left.\mathrm{1}\right)\:{decompose}\:{at}\:{simple}\:{elements} \\ $$$${U}_{{n}} =\:\frac{{n}\:{x}^{{n}−\mathrm{1}} }{{x}^{{n}} −\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{x}−{e}^{{it}} } \\ $$

Question Number 50388 Answers: 0 Comments: 0

find inf_((a,b)∈R^2 ) ∫_0 ^1 x^2 (ln(x)−ax−b)^2 dx

$${find}\:{inf}_{\left({a},{b}\right)\in{R}^{\mathrm{2}} } \:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} \left({ln}\left({x}\right)−{ax}−{b}\right)^{\mathrm{2}} {dx} \\ $$

Question Number 50384 Answers: 1 Comments: 1

find ∫ (dx/((1−x^2 )(1−x^3 ))) 2) calculate ∫_2 ^(√5) (dx/((1−x^2 )(1−x^3 )))

$${find}\:\int\:\:\:\frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$

Question Number 50219 Answers: 7 Comments: 0

Question Number 50423 Answers: 1 Comments: 1

find f(a) =∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0

$${find}\:{f}\left({a}\right)\:=\int_{{a}} ^{+\infty} \:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 50186 Answers: 1 Comments: 0

Let f(x)= ∫_2 ^( x) (dt/(1+t^6 )). Prove that : (1/(730))<f(3)<(1/(65)).

$${Let}\:{f}\left({x}\right)=\:\int_{\mathrm{2}} ^{\:{x}} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{6}} }. \\ $$$${Prove}\:{that}\:\::\:\frac{\mathrm{1}}{\mathrm{730}}<{f}\left(\mathrm{3}\right)<\frac{\mathrm{1}}{\mathrm{65}}. \\ $$

Question Number 50161 Answers: 1 Comments: 0

Find the function whose first derivative is 8−(5/(x^2 )^(1/3) ) the initial conditions f(8)=−20

$${Find}\:{the}\:{function}\:{whose}\:{first}\: \\ $$$${derivative}\:{is}\:\mathrm{8}−\frac{\mathrm{5}}{\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }}\:{the}\:{initial}\: \\ $$$${conditions}\:{f}\left(\mathrm{8}\right)=−\mathrm{20} \\ $$

Question Number 50158 Answers: 7 Comments: 1

Question Number 50052 Answers: 5 Comments: 1

Question Number 50007 Answers: 1 Comments: 1

For a < x < b, find ∫_a ^b (√(x−a)) . (√(b−x)) dx

$$\mathrm{For}\:{a}\:<\:{x}\:<\:{b},\:\mathrm{find}\: \\ $$$$\underset{{a}} {\overset{{b}} {\int}}\:\sqrt{{x}−{a}}\:.\:\sqrt{{b}−{x}}\:{dx} \\ $$

Question Number 49968 Answers: 0 Comments: 1

find f(x) =∫_0 ^(+∞) ((t arctan(xt))/(1+t^4 )) dt

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{t}\:{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$

Question Number 49967 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (dx/((x^2 −i)^2 ))

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

Question Number 49956 Answers: 1 Comments: 1

find ∫_0 ^1 cos(n arcosx)dx with n integr natural.

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left({n}\:{arcosx}\right){dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$

Question Number 49953 Answers: 0 Comments: 3

1) calculate ∫_0 ^1 ln(1+ix)dx and ∫_0 ^1 ln(1−ix)dx 2) find the value of ∫_0 ^1 ln(1+x^2 )dx .

$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ix}\right){dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx}\:. \\ $$

Question Number 49950 Answers: 1 Comments: 1

Question Number 49954 Answers: 1 Comments: 1

find f(α) =∫_0 ^1 ((arctan(αx))/(1+α^2 x^2 ))dx 2) calculate ∫_0 ^1 ((arctan(2x))/(1+4x^2 )) dx and ∫_0 ^1 ((arctan(3x))/(1+9x^2 )) dx .

$${find}\:\:\:{f}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+\alpha^{\mathrm{2}} {x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }\:\:{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{3}{x}\right)}{\mathrm{1}+\mathrm{9}{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

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