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IntegrationQuestion and Answers: Page 222

Question Number 68046 Answers: 1 Comments: 0

Question Number 68040 Answers: 1 Comments: 2

find f(a) =∫_1 ^2 arctan(x+(a/x))dx and calculate f^′ (a) at form of integral

$${find}\:{f}\left({a}\right)\:=\int_{\mathrm{1}} ^{\mathrm{2}} {arctan}\left({x}+\frac{{a}}{{x}}\right){dx}\:\:{and} \\ $$$${calculate}\:{f}^{'} \left({a}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$

Question Number 68039 Answers: 0 Comments: 1

find ∫ arctan(x+(1/x))dx

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

Question Number 68038 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((arctan(x^2 −1))/(x^2 +4))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} −\mathrm{1}\right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 68037 Answers: 1 Comments: 0

find ∫ ((x^2 dx)/((x^3 −8)(x^4 +1)))

$${find}\:\int\:\:\:\frac{{x}^{\mathrm{2}} {dx}}{\left({x}^{\mathrm{3}} −\mathrm{8}\right)\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)} \\ $$

Question Number 68033 Answers: 1 Comments: 0

find ∫ (dx/(1+sinx +sin(2x)))

$${find}\:\int\:\:\frac{{dx}}{\mathrm{1}+{sinx}\:+{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 67963 Answers: 1 Comments: 1

Question Number 67959 Answers: 0 Comments: 1

∫(√(e^y^2 )) dy pleas sir can you help me?

$$\int\sqrt{{e}^{{y}^{\mathrm{2}} } \:\:}\:{dy}\:\:{pleas}\:{sir}\:{can}\:{you}\:{help}\:{me}? \\ $$

Question Number 67942 Answers: 0 Comments: 1

∫e^(y^2 /2) dy

$$\int{e}^{{y}^{\mathrm{2}} /\mathrm{2}} \:\:{dy} \\ $$

Question Number 67937 Answers: 0 Comments: 1

Question Number 67932 Answers: 1 Comments: 4

let A(θ) = ∫_0 ^∞ (dx/((x^2 +3)(x^4 −e^(iθ) ))) with 0<θ<(π/2) 1) calculate A(θ) interms of θ 2) determine also ∫_0 ^∞ (dx/((x^2 +3)(x^4 −e^(iθ) )^2 ))

$${let}\:{A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta} \right)}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}\left(\theta\right)\:{interms}\:{of}\:\theta \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta} \right)^{\mathrm{2}} } \\ $$

Question Number 67931 Answers: 0 Comments: 0

let A_n =∫_0 ^(π/4) x^n {1+cosx +cos(2x)}^2 dx find a relation of recurrence betwedn the A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{x}^{{n}} \left\{\mathrm{1}+{cosx}\:+{cos}\left(\mathrm{2}{x}\right)\right\}^{\mathrm{2}} {dx} \\ $$$${find}\:{a}\:{relation}\:{of}\:{recurrence}\:{betwedn}\:{the}\:{A}_{{n}} \\ $$

Question Number 67919 Answers: 0 Comments: 0

Question Number 67921 Answers: 0 Comments: 0

∫e^(y^2 /2) dy

$$\int{e}^{{y}^{\mathrm{2}} /\mathrm{2}} {dy} \\ $$

Question Number 67907 Answers: 1 Comments: 1

Question Number 67851 Answers: 0 Comments: 5

find ∫ (dx/(x^2 −z)) with z from C .

$${find}\:\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} −{z}}\:\:{with}\:{z}\:{from}\:{C}\:. \\ $$

Question Number 67850 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/(x^2 −z)) with z from C

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{2}} −{z}}\:\:{with}\:{z}\:{from}\:{C} \\ $$

Question Number 67835 Answers: 1 Comments: 0

∫_0 ^2 x(8−x^3 )^(1/3) dx

$$\int_{\mathrm{0}} ^{\mathrm{2}} {x}\left(\mathrm{8}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx} \\ $$

Question Number 67823 Answers: 0 Comments: 0

∫_0 ^1 x^(lnx+e^(lnx/x) ) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{lnx}+{e}^{{lnx}/{x}} } {dx} \\ $$

Question Number 67799 Answers: 1 Comments: 3

calculate ∫_0 ^∞ (dx/((x^2 −e^(ia) )(x^2 −e^(ib) ))) with a>0 andb>0

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:−{e}^{{ia}} \right)\left({x}^{\mathrm{2}} −{e}^{{ib}} \right)}\:\:{with}\:{a}>\mathrm{0}\:{andb}>\mathrm{0} \\ $$

Question Number 67744 Answers: 0 Comments: 4

let f(x) =∫_0 ^∞ ((sin(t^2 ))/((x^2 +t^2 )^2 ))dt with x>0 1)determine a explicit form for f(x) 2) find also g(x) =∫_0 ^∞ ((sin(t^2 ))/((x^2 +t^2 )^3 ))dt 3) give f^((n)) (x) at form of integral and calculate f^((n)) (1). 4) find the valueof ∫_0 ^∞ ((sin(t^2 ))/((1+t^2 )^2 )) dt and ∫_0 ^∞ ((sin(t^2 ))/((1+t^2 )^3 ))dt

Question Number 67708 Answers: 1 Comments: 0

Question Number 67698 Answers: 0 Comments: 0

Question Number 67696 Answers: 0 Comments: 0

∫_0 ^1 (1−x)/(1+x)lnx dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}\right)/\left(\mathrm{1}+{x}\right){lnx}\:{dx} \\ $$

Question Number 67674 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a))) with a>0 1) determine a explicit form of f(a) 2) calculate g(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a)^2 )) 3)give f^((n)) (a) at form of integral 4)calculate ∫_0 ^∞ (dx/((x^2 +1)(x^2 +3)^2 )) and ∫_0 ^∞ (dx/((x^2 +1)^3 ))

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +{a}\right)}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({a}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 67673 Answers: 0 Comments: 1

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

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

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