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IntegrationQuestion and Answers: Page 100

Question Number 136396 Answers: 0 Comments: 1

find ∫_0 ^1 (x^a /(1−x))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{a}} }{\mathrm{1}−\mathrm{x}}\mathrm{dx} \\ $$

Question Number 136381 Answers: 1 Comments: 0

......sdvanced cslculus...... if x∈R^+ and:: 𝛗(x)=∫_0 ^( x) ((e^t −1)/t)ln((x/t))dt then prove that :: Ψ=∫_0 ^( ∞) e^(−x) 𝛗(x)dx=ζ(2)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:......{sdvanced}\:\:\:{cslculus}...... \\ $$$$\:{if}\:\:{x}\in\mathbb{R}^{+} \:{and}::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\left({x}\right)=\int_{\mathrm{0}} ^{\:{x}} \frac{{e}^{{t}} −\mathrm{1}}{{t}}{ln}\left(\frac{{x}}{{t}}\right){dt} \\ $$$$\:\:{then}\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}} \boldsymbol{\phi}\left({x}\right){dx}=\zeta\left(\mathrm{2}\right) \\ $$

Question Number 136365 Answers: 3 Comments: 0

∫ (dx/(sin x (√(cos x)))) =?

$$\int\:\frac{{dx}}{\mathrm{sin}\:{x}\:\sqrt{\mathrm{cos}\:{x}}}\:=? \\ $$

Question Number 136343 Answers: 1 Comments: 0

lim_(x→0) (x^(2021) /(x−ln(Σ_(k=0) ^(2020) (x^k /(k!))))) =^? 2021!

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{x}^{\mathrm{2021}} }{{x}−{ln}\left(\underset{{k}=\mathrm{0}} {\overset{\mathrm{2020}} {\sum}}\frac{{x}^{{k}} }{{k}!}\right)}\:\overset{?} {=}\:\mathrm{2021}!\: \\ $$

Question Number 136333 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (t^a /(1+t+t^2 ))dt study first the convergence (a real)

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}} }{\mathrm{1}+{t}+{t}^{\mathrm{2}} }{dt} \\ $$$${study}\:{first}\:{the}\:{convergence} \\ $$$$\left({a}\:{real}\right) \\ $$

Question Number 136325 Answers: 0 Comments: 1

please a generall Form for C(n) C(n)=(4/π^2 )Σ_(k=1) ^n (−1)^(k−1) ζ(2k)ζ(2n−2k)

$${please}\:{a}\:{generall}\:{Form}\:{for}\:{C}\left({n}\right) \\ $$$${C}\left({n}\right)=\frac{\mathrm{4}}{\pi^{\mathrm{2}} }\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \zeta\left(\mathrm{2}{k}\right)\zeta\left(\mathrm{2}{n}−\mathrm{2}{k}\right) \\ $$

Question Number 136279 Answers: 1 Comments: 0

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.

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

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