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IntegrationQuestion and Answers: Page 257

Question Number 49829    Answers: 1   Comments: 1

∫(x^2 /(x^4 +1))dx

$$\int\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} +\mathrm{1}}{dx} \\ $$

Question Number 49827    Answers: 1   Comments: 4

The integral ∫_0 ^(1/2) ((ln (1+2x))/(1+4x^2 ))dx = ? a) (π/4)ln2 b)(π/8)ln2 c)(π/(16))ln2 d)(π/(32))ln2

$${The}\:{integral}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }{dx}\:=\:? \\ $$$$\left.{a}\left.\right)\left.\:\left.\frac{\pi}{\mathrm{4}}{ln}\mathrm{2}\:\:\:\:{b}\right)\frac{\pi}{\mathrm{8}}{ln}\mathrm{2}\:\:\:\:{c}\right)\frac{\pi}{\mathrm{16}}{ln}\mathrm{2}\:\:\:{d}\right)\frac{\pi}{\mathrm{32}}{ln}\mathrm{2} \\ $$

Question Number 49816    Answers: 2   Comments: 0

((sin^6 x−cos^6 x)/(sin^2 xcos^2 x)).intregrate

$$\frac{\mathrm{sin}^{\mathrm{6}} \mathrm{x}−\mathrm{cos}^{\mathrm{6}} \mathrm{x}}{\mathrm{sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x}}.\mathrm{intregrate} \\ $$

Question Number 49815    Answers: 0   Comments: 3

sin^6 x−cos^6 x/sin^2 xcos^2 x

$$\mathrm{sin}^{\mathrm{6}} \mathrm{x}−\mathrm{cos}^{\mathrm{6}} \mathrm{x}/\mathrm{sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x} \\ $$

Question Number 49806    Answers: 0   Comments: 1

let f(x) =∫_0 ^(π/4) ln(1−x^2 cosθ)dθ with ∣x∣<1 1) find a explicit form of f(x) 2) calculate ∫_0 ^(π/4) ln(1−(1/4)cosθ)dθ .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} {cos}\theta\right){d}\theta\:\:\:{with}\:\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}{cos}\theta\right){d}\theta\:. \\ $$

Question Number 49761    Answers: 1   Comments: 0

Calculate : ∫(( sin^2 x cos^2 x)/((sin^3 x+cos^3 x)^2 )) dx

$${Calculate}\:: \\ $$$$\:\int\frac{\:\:\mathrm{sin}^{\mathrm{2}} {x}\:\mathrm{cos}^{\mathrm{2}} {x}}{\left(\mathrm{sin}^{\mathrm{3}} {x}+\mathrm{cos}^{\mathrm{3}} {x}\right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 49760    Answers: 0   Comments: 3

Question Number 49746    Answers: 1   Comments: 0

∫((sin^8 x−cos^8 x)/(1−2sin^2 x.cos^2 x)) = ? a) ((−1)/2)sin 2x b)(1/2)sin 2x c)None.

$$\int\frac{\mathrm{sin}^{\mathrm{8}} {x}−\mathrm{cos}^{\mathrm{8}} {x}}{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} {x}.\mathrm{cos}^{\mathrm{2}} {x}}\:=\:? \\ $$$$\left.{a}\left.\right)\left.\:\frac{−\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}\:\:\:{b}\right)\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}\:\:\:{c}\right){None}. \\ $$

Question Number 49708    Answers: 0   Comments: 0

Please integrate ∫(((e^(cos x) sin x)/(1−x^2 )))dx

$${Please}\:{integrate} \\ $$$$\int\left(\frac{\mathrm{e}^{\mathrm{cos}\:{x}} \mathrm{sin}\:{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 49661    Answers: 1   Comments: 2

calculateA_n =(1/(2i)) ∫_0 ^1 {(1+ix)^n −(1−ix)^n }dx

$${calculateA}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}{i}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left\{\left(\mathrm{1}+{ix}\right)^{{n}} −\left(\mathrm{1}−{ix}\right)^{{n}} \right\}{dx} \\ $$

Question Number 49646    Answers: 0   Comments: 0

calculate ∫∫_D (x^2 −y^2 )(√(x^2 +y^2 ))dxdy with D ={(x,y)∈R^2 / −1≤x≤1 and 0≤y≤2 }

$${calculate}\:\int\int_{{D}} \left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{2}\:\right\} \\ $$

Question Number 49645    Answers: 1   Comments: 0

calculate ∫∫_C ∣x+y∣dxdy with C=[−1,1]×[−1,1]

$${calculate}\:\int\int_{{C}} \:\mid{x}+{y}\mid{dxdy}\:\:{with}\:{C}=\left[−\mathrm{1},\mathrm{1}\right]×\left[−\mathrm{1},\mathrm{1}\right] \\ $$

Question Number 49636    Answers: 1   Comments: 2

1) calculate A_n =∫_0 ^∞ e^(−n[x]) sin(x)dx with n integr and n≥1 2) find nature of Σ_(n=1) ^∞ A_n

$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{n}\left[{x}\right]} {sin}\left({x}\right){dx}\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{A}_{{n}} \\ $$

Question Number 49635    Answers: 1   Comments: 1

1)find f(x) =∫_0 ^(π/4) ((sint)/(2+x cos(2t)))dt 2) find g(x) =∫_0 ^(π/4) ((sint sin(2t)/((2+x cos(2t))^2 ))dx 3) find the value of ∫_0 ^(π/4) ((sint)/(2+3 cos(2t)))dt and ∫_0 ^(π/4) ((sin(t)sin(2t))/((2+3cos(2t))^2 ))dt

$$\left.\mathrm{1}\right){find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{sint}}{\mathrm{2}+{x}\:{cos}\left(\mathrm{2}{t}\right)}{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{sint}\:{sin}\left(\mathrm{2}{t}\right.}{\left(\mathrm{2}+{x}\:{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{sint}}{\mathrm{2}+\mathrm{3}\:{cos}\left(\mathrm{2}{t}\right)}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{sin}\left({t}\right){sin}\left(\mathrm{2}{t}\right)}{\left(\mathrm{2}+\mathrm{3}{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 49457    Answers: 0   Comments: 0

evaluate ∫x^(3 ) J_3 (x)dx

$${evaluate}\:\int{x}^{\mathrm{3}\:} {J}_{\mathrm{3}} \left({x}\right){dx} \\ $$

Question Number 49392    Answers: 1   Comments: 0

Question Number 49367    Answers: 5   Comments: 5

a) ∫ (dx/(√(1−tgx))) b)∫ (dx/((1−tgx))^(1/3) ) c)∫ (dx/(√(1−(√(1−x)))))

$$\left.\:\:\:\:{a}\right)\:\:\int\:\:\frac{\boldsymbol{\mathrm{dx}}}{\sqrt{\mathrm{1}−\boldsymbol{\mathrm{tgx}}}} \\ $$$$\left.\:\:\:\:{b}\right)\int\:\:\frac{\boldsymbol{\mathrm{dx}}}{\sqrt[{\mathrm{3}}]{\mathrm{1}−\boldsymbol{\mathrm{tgx}}}} \\ $$$$\left.\:\:\:\:{c}\right)\int\:\:\frac{\boldsymbol{\mathrm{dx}}}{\sqrt{\mathrm{1}−\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}}}} \\ $$

Question Number 49344    Answers: 1   Comments: 1

let α>0 calculate ∫_(−∞) ^(+∞) (1+αi)^(−x^2 ) dx .

$${let}\:\alpha>\mathrm{0}\:{calculate}\:\int_{−\infty} ^{+\infty} \:\left(\mathrm{1}+\alpha{i}\right)^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 49343    Answers: 0   Comments: 1

find ∫_0 ^1 ((ln(x))/(1+x))dx .

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

Question Number 49342    Answers: 0   Comments: 0

find ∫_0 ^1 (e^x /(1+x))dx .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{{x}} }{\mathrm{1}+{x}}{dx}\:. \\ $$

Question Number 49252    Answers: 1   Comments: 0

Question Number 49250    Answers: 1   Comments: 0

Apply derivative criteria F(x)=x^3 +5x^2 −2x+3

$${Apply}\:{derivative}\:{criteria} \\ $$$${F}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{3} \\ $$

Question Number 49232    Answers: 0   Comments: 2

let f(a)= ∫_(−∞) ^(+∞) cos(x^2 +ax +1)dx 1)calculate f(a) and f^′ (a) 2) find f^((n)) (a)

$${let}\:{f}\left({a}\right)=\:\int_{−\infty} ^{+\infty} {cos}\left({x}^{\mathrm{2}} \:+{ax}\:+\mathrm{1}\right){dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}\left({a}\right)\:{and}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{f}^{\left({n}\right)} \left({a}\right)\: \\ $$

Question Number 49225    Answers: 1   Comments: 0

∫(1/(z(z^(17) +1)))=?? find please

$$\int\frac{\mathrm{1}}{{z}\left({z}^{\mathrm{17}} +\mathrm{1}\right)}=??\:{find}\:{please} \\ $$

Question Number 49187    Answers: 2   Comments: 0

∫((sinx)/(sin4x))dx

$$\int\frac{{sinx}}{{sin}\mathrm{4}{x}}{dx} \\ $$$$ \\ $$

Question Number 49144    Answers: 0   Comments: 2

∫(1/(x^n +1))dx=??

$$\int\frac{\mathrm{1}}{{x}^{{n}} +\mathrm{1}}{dx}=?? \\ $$

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