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IntegrationQuestion and Answers: Page 101

Question Number 136267    Answers: 0   Comments: 1

Question Number 136211    Answers: 1   Comments: 0

hi, guyz ! please, i need ur help ! I=∫_1 ^( 5) xe^x^3 dx

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{guyz}}\:!\: \\ $$$$\boldsymbol{\mathrm{please}},\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{ur}}\:\boldsymbol{\mathrm{help}}\:! \\ $$$$\boldsymbol{\mathrm{I}}=\int_{\mathrm{1}} ^{\:\mathrm{5}} \boldsymbol{{xe}}^{\boldsymbol{{x}}^{\mathrm{3}} } \boldsymbol{{dx}} \\ $$

Question Number 136210    Answers: 5   Comments: 3

Show that ∫_0 ^∞ ((lnx)/((x^2 +1)^2 ))dx=−(π/4)

$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{lnx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx}=−\frac{\pi}{\mathrm{4}} \\ $$

Question Number 136197    Answers: 1   Comments: 0

a. Prove that for any real constant a ∫_0 ^∞ e^(−(a/x^2 )) dx=∞ b. If a and b are real constants, explain why we cannot split the integral ∫_0 ^∞ (e^(−(a/x^2 )) −e^(−(b/x^2 )) )dx as the difference ∫_0 ^∞ e^(−(a/x^2 )) dx−∫_0 ^∞ e^(−(b/x^2 )) dx c. If a≥0 and b≥0 constants, then prove that ∫_0 ^∞ (e^(−(a/x^2 )) −e^(−(b/x^2 )) )dx=(√(πb))−(√(πa)). d. If a>b≥0 constants, then prove that ∫_0 ^∞ (e^(−(a/x^2 )) −e^(−(b/x^2 )) )dx=∞

$$\mathrm{a}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{real}\:\mathrm{constant}\:\mathrm{a}\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx}=\infty \\ $$$$\mathrm{b}.\:\mathrm{If}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{real}\:\mathrm{constants},\:\mathrm{explain}\:\mathrm{why}\:\mathrm{we}\:\mathrm{cannot}\:\mathrm{split}\:\mathrm{the} \\ $$$$\mathrm{integral}\:\:\int_{\mathrm{0}} ^{\infty} \left(\mathrm{e}^{−\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }} −\mathrm{e}^{−\frac{\mathrm{b}}{\mathrm{x}^{\mathrm{2}} }} \right)\mathrm{dx}\:\mathrm{as}\:\mathrm{the}\:\mathrm{difference}\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx}−\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\frac{\mathrm{b}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx} \\ $$$$\mathrm{c}.\:\mathrm{If}\:\mathrm{a}\geqslant\mathrm{0}\:\mathrm{and}\:\mathrm{b}\geqslant\mathrm{0}\:\mathrm{constants},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{e}^{−\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }} −\mathrm{e}^{−\frac{\mathrm{b}}{\mathrm{x}^{\mathrm{2}} }} \right)\mathrm{dx}=\sqrt{\pi\mathrm{b}}−\sqrt{\pi\mathrm{a}}. \\ $$$$\mathrm{d}.\:\mathrm{If}\:\mathrm{a}>\mathrm{b}\geqslant\mathrm{0}\:\mathrm{constants},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \left(\mathrm{e}^{−\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }} −\mathrm{e}^{−\frac{\mathrm{b}}{\mathrm{x}^{\mathrm{2}} }} \right)\mathrm{dx}=\infty \\ $$

Question Number 136170    Answers: 1   Comments: 0

.....nice calculus.... compute:: 2li_2 (((−1)/2))−2li_2 ((1/2))+li_2 ((3/4))=??

$$\:\:\:\:\:\:\:\:\:\:\:\:.....{nice}\:\:{calculus}.... \\ $$$$\:\:\:{compute}:: \\ $$$$\mathrm{2}{li}_{\mathrm{2}} \left(\frac{−\mathrm{1}}{\mathrm{2}}\right)−\mathrm{2}{li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)+{li}_{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)=?? \\ $$$$ \\ $$

Question Number 136132    Answers: 1   Comments: 0

Find a series for (x^2 /(tanh (xπ)tan (xπ)))

$${Find}\:{a}\:{series}\:{for}\:\frac{{x}^{\mathrm{2}} }{\mathrm{tanh}\:\left({x}\pi\right)\mathrm{tan}\:\left({x}\pi\right)} \\ $$

Question Number 136076    Answers: 1   Comments: 0

find the area between the curve y = 3 + 2x −x^2 , the x−axis and the line y = 3. find the volume of the solid generated when the curve is rotated completely about the line y = 3

$$\mathrm{find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{between}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:\mathrm{3}\:+\:\mathrm{2}{x}\:−{x}^{\mathrm{2}} ,\:\mathrm{the}\:{x}−\mathrm{axis} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{line}\:{y}\:=\:\mathrm{3}. \\ $$$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{generated}\:\mathrm{when}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{is} \\ $$$$\mathrm{rotated}\:\mathrm{completely}\:\mathrm{about}\:\mathrm{the}\:\mathrm{line}\:{y}\:=\:\mathrm{3} \\ $$

Question Number 136067    Answers: 2   Comments: 0

Λ = ∫ x^3 (x^3 +1)^(10) dx

$$\Lambda\:=\:\int\:{x}^{\mathrm{3}} \:\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{10}} \:{dx}\: \\ $$

Question Number 136060    Answers: 1   Comments: 0

...advanced calculus.... evaluate:: 𝛗=Im(∫_0 ^( (π/2)) li_2 (sin(x))+li_2 (csc(x))dx)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:\:{calculus}.... \\ $$$$\:\:\:\:{evaluate}:: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\mathrm{Im}\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {li}_{\mathrm{2}} \left({sin}\left({x}\right)\right)+{li}_{\mathrm{2}} \left({csc}\left({x}\right)\right){dx}\right) \\ $$$$ \\ $$

Question Number 136036    Answers: 1   Comments: 4

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

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 136034    Answers: 1   Comments: 0

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

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

Question Number 136023    Answers: 0   Comments: 1

calculate ∫_0 ^∞ e^(−z^2 ) dz with z complex

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{z}^{\mathrm{2}} } \mathrm{dz}\:\:\mathrm{with}\:\mathrm{z}\:\mathrm{complex} \\ $$

Question Number 135996    Answers: 1   Comments: 0

Question Number 135957    Answers: 2   Comments: 0

1) find ∫ (dx/((x+1)^2 (x−3)^4 )) 2) deduce the decomposition of F(x)=(1/((x+1)^2 (x−3)^4 ))

$$\left.\mathrm{1}\right)\:\mathrm{find}\:\int\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{decomposition}\:\mathrm{of}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$

Question Number 135932    Answers: 2   Comments: 0

Evaluate (1) ∫_0 ^1 ∫_0 ^x ∫_0 ^y (3x^2 +2y^2 −3z^2 )dxdydz (2) ∫(2x−2)^3 dx (3) ∫(((x−5)/(x^2 −10x+2)))dx

$${Evaluate}\:\left(\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{{x}} \int_{\mathrm{0}} ^{{y}} \left(\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} −\mathrm{3}{z}^{\mathrm{2}} \right){dxdydz} \\ $$$$\left(\mathrm{2}\right)\:\int\left(\mathrm{2}{x}−\mathrm{2}\right)^{\mathrm{3}} {dx} \\ $$$$\left(\mathrm{3}\right)\:\int\left(\frac{{x}−\mathrm{5}}{{x}^{\mathrm{2}} −\mathrm{10}{x}+\mathrm{2}}\right){dx} \\ $$

Question Number 135888    Answers: 1   Comments: 0

Question Number 135872    Answers: 1   Comments: 0

Evaluate ∮_c ydy where c is a circle x^2 +y^2 =4

$${Evaluate}\:\oint_{{c}} {ydy}\:{where}\:\:{c}\:{is}\:{a}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{4} \\ $$

Question Number 135867    Answers: 3   Comments: 0

sin^2 (4x)+cos^2 (x)=2sin (4x)cos^2 (x)

$$\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{4}{x}\right)+\mathrm{cos}\:^{\mathrm{2}} \left({x}\right)=\mathrm{2sin}\:\left(\mathrm{4}{x}\right)\mathrm{cos}\:^{\mathrm{2}} \left({x}\right) \\ $$$$ \\ $$

Question Number 135820    Answers: 1   Comments: 0

Let f(x)=∫_0 ^x e^(−t^2 ) dt , Prove ∫_0 ^∞ e^(−x^2 +f(x)) dx=e^((√π)/2) −1.

$$\mathrm{Let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {e}^{−{t}^{\mathrm{2}} } {dt}\:, \\ $$$$\mathrm{Prove}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{x}^{\mathrm{2}} +{f}\left({x}\right)} {dx}={e}^{\frac{\sqrt{\pi}}{\mathrm{2}}} −\mathrm{1}. \\ $$

Question Number 135784    Answers: 1   Comments: 0

Given { ((f(3)=4 , f ′(3)=−2)),((f(8)=5 , f ′(8)=3)) :} find ∫_3 ^( 8) x f ′′(x) dx .

$${Given}\:\begin{cases}{{f}\left(\mathrm{3}\right)=\mathrm{4}\:,\:{f}\:'\left(\mathrm{3}\right)=−\mathrm{2}}\\{{f}\left(\mathrm{8}\right)=\mathrm{5}\:,\:{f}\:'\left(\mathrm{8}\right)=\mathrm{3}}\end{cases} \\ $$$${find}\:\int_{\mathrm{3}} ^{\:\mathrm{8}} \:{x}\:{f}\:''\left({x}\right)\:{dx}\:. \\ $$

Question Number 135753    Answers: 1   Comments: 1

Question Number 135777    Answers: 1   Comments: 0

let U_n =∫_(−∞) ^(+∞) ((cos(nx))/((x^2 −x+1)^2 ))dx calculate lim_(n→+∞) e^n^2 U_n

$$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{nx}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx}\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{e}^{\mathrm{n}^{\mathrm{2}} } \mathrm{U}_{\mathrm{n}} \\ $$

Question Number 135737    Answers: 2   Comments: 0

∫_(−2π) ^(4π) (3/(5−4cosx))dx

$$\int_{−\mathrm{2}\pi} ^{\mathrm{4}\pi} \frac{\mathrm{3}}{\mathrm{5}−\mathrm{4cosx}}\mathrm{dx} \\ $$

Question Number 135697    Answers: 0   Comments: 0

Question Number 135693    Answers: 1   Comments: 0

Ω = ∫ ((x−1)/((x−2)(x^2 −2x+2)^2 )) dx

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

Question Number 135673    Answers: 2   Comments: 0

∫ ((x^2 +1)/(x^4 +x^2 +1)) dx

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

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