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IntegrationQuestion and Answers: Page 20

Question Number 204910 Answers: 1 Comments: 1

Question Number 204902 Answers: 2 Comments: 0

calculate ∫_0 ^1 (√(x(1−x)))dx

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

Question Number 204901 Answers: 0 Comments: 0

Question Number 204866 Answers: 1 Comments: 0

∫ ((x+3)/(x^2 (√(2x+3)))) dx=?

$$\int\:\frac{{x}+\mathrm{3}}{{x}^{\mathrm{2}} \sqrt{\mathrm{2}{x}+\mathrm{3}}}\:{dx}=? \\ $$

Question Number 204802 Answers: 1 Comments: 0

Wi-Fi code problem: ∫_(−2) ^( 2) (x^3 cos((x/2))+(1/2))(√(4−x^2 ))dx

$$\mathrm{Wi}-\mathrm{Fi}\:\mathrm{code}\:\mathrm{problem}: \\ $$$$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \left({x}^{\mathrm{3}} \mathrm{cos}\left(\frac{{x}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\mathrm{d}{x} \\ $$

Question Number 204707 Answers: 1 Comments: 0

integrate ∫_0 ^∞ (e^(−x^2 ) /(1+e^x ))dx

$$\boldsymbol{{integrate}}\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} } }{\mathrm{1}+\boldsymbol{{e}}^{\boldsymbol{{x}}} }\boldsymbol{{dx}} \\ $$

Question Number 204706 Answers: 0 Comments: 0

evaluate ∫_0 ^∞ 2^(−𝚪(x)) dx

$$\boldsymbol{{evaluate}}\:\int_{\mathrm{0}} ^{\infty} \mathrm{2}^{−\boldsymbol{\Gamma}\left(\boldsymbol{{x}}\right)} \boldsymbol{{dx}} \\ $$

Question Number 204705 Answers: 0 Comments: 1

evalute ∫_0 ^∞ 2^(−(√(tanx))) dx

$$\boldsymbol{{evalute}}\:\int_{\mathrm{0}} ^{\infty} \mathrm{2}^{−\sqrt{\boldsymbol{{tanx}}}} \boldsymbol{{dx}} \\ $$

Question Number 204645 Answers: 1 Comments: 1

Let f : [ 1^ ∞) →R be a differentiable function such that f(1)= (1/3) and 3∫_1 ^x f(t) dt = x f(x)−(x^3 /3) ,x∈[1,∞) find tbe value of f(e)

$$\:\:\mathrm{Let}\:{f}\::\:\left[\:\bar {\mathrm{1}}\infty\right)\:\rightarrow\mathrm{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{differentiable}\: \\ $$$$\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{1}\right)=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{and}\: \\ $$$$\:\mathrm{3}\underset{\mathrm{1}} {\overset{\mathrm{x}} {\int}}\:{f}\left({t}\right)\:{dt}\:=\:{x}\:{f}\left({x}\right)−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:,\mathrm{x}\in\left[\mathrm{1},\infty\right)\: \\ $$$$\:\mathrm{find}\:\mathrm{tbe}\:\mathrm{value}\:\mathrm{of}\:{f}\left({e}\right)\: \\ $$

Question Number 204573 Answers: 0 Comments: 3

How Can derive LambertW(z) in the Form of integral??? W(z)=(1/π)∫_0 ^( π) ln(1+((z∙sin(t))/t)e^(t∙cot(t)) )dt , z∈[−(1/e),∞) Or Similar to the example.LambertW(z) How other Functions can be Derived in Integral Form

$$\mathrm{How}\:\mathrm{Can}\:\mathrm{derive}\:\mathrm{LambertW}\left({z}\right)\:\mathrm{in}\:\mathrm{the} \\ $$$$\:\mathrm{Form}\:\mathrm{of}\:\mathrm{integral}??? \\ $$$$\mathrm{W}\left({z}\right)=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:\mathrm{ln}\left(\mathrm{1}+\frac{{z}\centerdot\mathrm{sin}\left({t}\right)}{{t}}{e}^{{t}\centerdot\mathrm{cot}\left({t}\right)} \right)\mathrm{d}{t}\:,\:{z}\in\left[−\frac{\mathrm{1}}{{e}},\infty\right) \\ $$$$\mathrm{Or}\:\mathrm{Similar}\:\mathrm{to}\:\mathrm{the}\:\mathrm{example}.\mathrm{LambertW}\left({z}\right) \\ $$$$\mathrm{How}\:\mathrm{other}\:\mathrm{Functions}\:\mathrm{can}\:\mathrm{be}\:\mathrm{Derived}\:\mathrm{in}\:\mathrm{Integral}\:\mathrm{Form} \\ $$

Question Number 204569 Answers: 1 Comments: 0

find the value of I=∫_0 ^(+∞) ln(1+e^(−x) )dx nowing that Σ_(n=1) ^(+∞) (1/n^2 )=(π^2 /6)

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\: \\ $$$$\boldsymbol{{I}}=\int_{\mathrm{0}} ^{+\infty} \boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{e}}^{−\boldsymbol{{x}}} \right)\boldsymbol{{dx}}\:\boldsymbol{{nowing}}\:\boldsymbol{{that}}\: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$

Question Number 204533 Answers: 1 Comments: 0

Question Number 204522 Answers: 1 Comments: 3

Question Number 204517 Answers: 2 Comments: 0

Question Number 204472 Answers: 2 Comments: 0

Calculate ... Ω=Σ_(k=1) ^n ⌊(( 1)/( (e)^(1/k) −1)) ⌋ =?

$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Calculate}\:... \\ $$$$\:\:\:\:\:\:\:\Omega=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\lfloor\frac{\:\mathrm{1}}{\:\sqrt[{{k}}]{{e}}\:−\mathrm{1}}\:\rfloor\:=? \\ $$$$ \\ $$

Question Number 204409 Answers: 3 Comments: 1

find ⌊∫_0 ^(2023) (2/(x+e^x ))dx⌋=?

$${find}\:\lfloor\int_{\mathrm{0}} ^{\mathrm{2023}} \frac{\mathrm{2}}{{x}+{e}^{{x}} }{dx}\rfloor=? \\ $$

Question Number 204275 Answers: 1 Comments: 0

Show that ∫_0 ^(π/4) (√(tan x)) (√(1−tan x)) dx=(((√((√2)−1))/( (√2)))−1)π

$$\mathrm{Show}\:\mathrm{that} \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\sqrt{\mathrm{tan}\:{x}}\:\sqrt{\mathrm{1}−\mathrm{tan}\:{x}}\:{dx}=\left(\frac{\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}{\:\sqrt{\mathrm{2}}}−\mathrm{1}\right)\pi \\ $$

Question Number 204244 Answers: 2 Comments: 0

Question Number 204233 Answers: 2 Comments: 0

Question Number 203964 Answers: 2 Comments: 0

Advanced calculus ... Q: If , ∫_0 ^1 ∫_0 ^( 1) (( xln(x)ln^2 (y ))/(1−xy)) dxdy = λ Σ_(n=1) ^∞ (( 1)/(n^( 3) ( n+1 )^2 )) ⇒ Find , λ =?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Advanced}\:\:{calculus}\:... \\ $$$$\:\:{Q}:\:\:\:{If}\:,\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\mathrm{ln}\left({x}\right)\mathrm{ln}^{\mathrm{2}} \left({y}\:\right)}{\mathrm{1}−{xy}}\:{dxdy}\:=\:\lambda\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:\mathrm{3}} \left(\:{n}+\mathrm{1}\:\right)^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{Find}\:,\:\:\:\:\:\lambda\:=? \\ $$$$ \\ $$

Question Number 203884 Answers: 1 Comments: 0

find ∫_0 ^∞ (x^3 /((1+x)^4 (x+2)^5 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }{dx} \\ $$

Question Number 203867 Answers: 1 Comments: 0

find ∫(√((1−x^3 )/(1+x^3 )))dx

$${find}\:\int\sqrt{\frac{\mathrm{1}−{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{3}} }}{dx} \\ $$

Question Number 203836 Answers: 2 Comments: 0

Question Number 203772 Answers: 2 Comments: 0

Question Number 203747 Answers: 2 Comments: 0

Question Number 203679 Answers: 0 Comments: 0

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