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IntegrationQuestion and Answers: Page 154

Question Number 100114    Answers: 1   Comments: 0

Question Number 100089    Answers: 0   Comments: 0

calculate A_n =∫_0 ^(π/2) ((sin^n (x))/(sin(nx)))dx

$$\:\mathrm{calculate}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{sin}^{\mathrm{n}} \left(\mathrm{x}\right)}{\mathrm{sin}\left(\mathrm{nx}\right)}\mathrm{dx}\: \\ $$

Question Number 100088    Answers: 1   Comments: 1

calculate ∫ ((cosx)/(cos(3x)))dx

$$\mathrm{calculate}\:\int\:\frac{\mathrm{cosx}}{\mathrm{cos}\left(\mathrm{3x}\right)}\mathrm{dx} \\ $$

Question Number 100054    Answers: 1   Comments: 0

I_(n,m) =∫_0 ^1 ∫_0 ^1 (((ln(x))^n (ln(y))^m )/(1−xy))dx dy

$${I}_{{n},{m}} =\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{{n}} \left({ln}\left({y}\right)\right)^{{m}} }{\mathrm{1}−{xy}}{dx}\:{dy} \\ $$

Question Number 100047    Answers: 2   Comments: 0

∫_0 ^1 e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$

Question Number 100026    Answers: 1   Comments: 0

∫_0 ^(π/2) e^(−sec^2 θ) dθ

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{e}^{−\mathrm{sec}^{\mathrm{2}} \theta} \mathrm{d}\theta \\ $$

Question Number 99831    Answers: 0   Comments: 0

solve the ds { ((x^′ +2y^′ =sint)),((3x^′ +y^′ =te^t )) :}

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{ds}\:\:\:\begin{cases}{\mathrm{x}^{'} \:+\mathrm{2y}^{'} \:=\mathrm{sint}}\\{\mathrm{3x}^{'} +\mathrm{y}^{'} \:=\mathrm{te}^{\mathrm{t}} }\end{cases} \\ $$

Question Number 99824    Answers: 1   Comments: 0

calculate ∫_0 ^1 xe^(−x^2 ) arctan((2/x))dx

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

Question Number 99820    Answers: 0   Comments: 0

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

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

Question Number 99818    Answers: 2   Comments: 0

∫_0 ^1 ln(1+(1/(n^2 x^2 )))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} }\right)\mathrm{dx} \\ $$

Question Number 99779    Answers: 1   Comments: 2

Question Number 99707    Answers: 4   Comments: 1

∫_(−∞) ^∞ e^(−x^2 ) dx=?

$$\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx}=? \\ $$

Question Number 99679    Answers: 0   Comments: 1

Question Number 99578    Answers: 2   Comments: 0

1)calculate U_n =∫_0 ^∞ e^(−nx^4 ) dx and determine lim_(n→+∞) n^4 U_n 2) find nature of the serie Σ U_n

$$\left.\mathrm{1}\right)\mathrm{calculate}\:\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{nx}^{\mathrm{4}} } \mathrm{dx}\:\:\mathrm{and}\:\mathrm{determine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{n}^{\mathrm{4}} \:\mathrm{U}_{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{serie}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 99576    Answers: 2   Comments: 0

1)let f(a) =∫_0 ^∞ (( t^(a−1) ln(t))/(1+t)) dt with 0<a<1 prove that f(a)is convergent and determine it value 2)calculate∫_0 ^∞ ((lnt)/((1+t)(√t)))dt 3)calculate∫_0 ^∞ ((lnt)/((^3 (√t^2 ))(1+t)))dt

$$\left.\mathrm{1}\right)\mathrm{let}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\:\mathrm{t}^{\mathrm{a}−\mathrm{1}} \mathrm{ln}\left(\mathrm{t}\right)}{\mathrm{1}+\mathrm{t}}\:\mathrm{dt}\:\:\:\mathrm{with}\:\mathrm{0}<\mathrm{a}<\mathrm{1}\:\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{a}\right)\mathrm{is}\:\mathrm{convergent}\:\mathrm{and}\:\mathrm{determine} \\ $$$$\mathrm{it}\:\mathrm{value} \\ $$$$\left.\mathrm{2}\right)\mathrm{calculate}\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnt}}{\left(\mathrm{1}+\mathrm{t}\right)\sqrt{\mathrm{t}}}\mathrm{dt} \\ $$$$\left.\mathrm{3}\right)\mathrm{calculate}\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnt}}{\left(^{\mathrm{3}} \sqrt{\mathrm{t}^{\mathrm{2}} }\right)\left(\mathrm{1}+\mathrm{t}\right)}\mathrm{dt} \\ $$

Question Number 99557    Answers: 1   Comments: 0

Λ=∫_0 ^1 (Li_2 ((1/(1+x)))+Li_3 ((x/(1+x)))+Li_4 (((x+1)/(x^2 +x+1))))dx Li_n (z)=polylogarithm function. by adeyemi.

$$ \\ $$$$\Lambda=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{Li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}}\right)+\mathrm{Li}_{\mathrm{3}} \left(\frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}}\right)+\mathrm{Li}_{\mathrm{4}} \left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\right)\right)\mathrm{dx} \\ $$$$\mathrm{Li}_{\mathrm{n}} \left(\mathrm{z}\right)=\mathrm{polylogarithm}\:\mathrm{function}. \\ $$$$\mathrm{by}\:\mathrm{adeyemi}. \\ $$$$ \\ $$

Question Number 99541    Answers: 3   Comments: 0

∫_(−∞) ^∞ e^(−2x^2 −5x−3) dx=? help me

$$\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}−\mathrm{3}} \mathrm{dx}=?\: \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$

Question Number 99536    Answers: 0   Comments: 0

∫tan^(πi) xdx

$$\int{tan}^{\pi{i}} {xdx} \\ $$

Question Number 99516    Answers: 1   Comments: 0

Question Number 99504    Answers: 4   Comments: 0

Question Number 99496    Answers: 1   Comments: 0

convergence radius of Σ_(n∈N) 2^n z^(n!)

$${convergence}\:{radius}\:{of}\:\:\underset{{n}\in\mathbb{N}} {\sum}\:\mathrm{2}^{{n}} {z}^{{n}!} \: \\ $$

Question Number 99421    Answers: 0   Comments: 0

Question Number 99413    Answers: 1   Comments: 0

∫_0 ^(+∞) ((sin(ax))/(e^(2πx) −1))dx

$$\int_{\mathrm{0}} ^{+\infty} \frac{{sin}\left({ax}\right)}{{e}^{\mathrm{2}\pi{x}} −\mathrm{1}}{dx} \\ $$

Question Number 99403    Answers: 1   Comments: 1

∫ x^x dx

$$\int\:\mathrm{x}^{\mathrm{x}} \:\:\mathrm{dx} \\ $$

Question Number 99326    Answers: 1   Comments: 0

Question Number 99315    Answers: 0   Comments: 0

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