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IntegrationQuestion and Answers: Page 77

Question Number 143163 Answers: 0 Comments: 0

∫_(1/x) ^x^2 (dt/( (√(1+t^3 )))) =?

$$\int_{\frac{\mathrm{1}}{{x}}} ^{{x}^{\mathrm{2}} } \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{3}} }}\:=? \\ $$

Question Number 143142 Answers: 1 Comments: 0

∫((x^2 −1)/(x^2 +1))∙(1/( (√(1+x^4 ))))dx

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

Question Number 143098 Answers: 1 Comments: 0

.....Prove.... Σ_(n=1) ^∞ ((1/(sinh(πn))))^2 =(1/6) −(1/(2π)) ... ......

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{Prove}....\: \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{s}{inh}\left(\pi{n}\right)}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{6}}\:−\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:... \\ $$$$\:\:\:\:\:\:\:...... \\ $$

Question Number 143094 Answers: 1 Comments: 0

Question Number 143087 Answers: 2 Comments: 0

Question Number 143083 Answers: 1 Comments: 0

calculate Ψ(a,b)=∫_0 ^∞ (e^(−ax^2 ) /((x^2 +b^2 )^2 ))dx with a>0 and b>0

$${calculate}\:\Psi\left({a},{b}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 143082 Answers: 2 Comments: 0

calculate f(a,b)=∫_0 ^∞ (e^(−ax^2 ) /(x^2 +b^2 ))dx with a>0 and b>0

$${calculate}\:{f}\left({a},{b}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}^{\mathrm{2}} } }{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }{dx} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 143081 Answers: 2 Comments: 0

calculate ∫_0 ^∞ xe^(−x^2 ) arctanx dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} {xe}^{−{x}^{\mathrm{2}} } {arctanx}\:{dx} \\ $$

Question Number 143080 Answers: 2 Comments: 0

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

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

Question Number 143071 Answers: 2 Comments: 0

∫_0 ^(π/4) ((8dx)/(tgx+1))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{8}{dx}}{{tgx}+\mathrm{1}} \\ $$

Question Number 143051 Answers: 1 Comments: 0

_(∗∗∗∗∗) :: Lobachevsky Integral ::_(∗∗∗∗∗) 𝛗:=∫_0 ^( ∞) ((sin^2 ( tan(x)))/x^( 2) )dx=^? (π/2) ..........

$$\:\:\:\:\:\:\:_{\ast\ast\ast\ast\ast} ::\:\:{Lobachevsky}\:{Integral}\:::_{\ast\ast\ast\ast\ast} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{s}{in}^{\mathrm{2}} \left(\:{tan}\left({x}\right)\right)}{{x}^{\:\mathrm{2}} }{dx}\overset{?} {=}\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:.......... \\ $$

Question Number 142990 Answers: 2 Comments: 0

find ∫_0 ^∞ (e^(−x^2 ) /((3+x^2 )^2 ))dx

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

Question Number 142989 Answers: 2 Comments: 0

calculate ∫_0 ^∞ (e^(−3x^2 ) /(1+x^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{e}^{−\mathrm{3x}^{\mathrm{2}} } }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 142983 Answers: 0 Comments: 0

Question Number 142917 Answers: 1 Comments: 0

∫(sin^7 (x))dx

$$\int\left(\boldsymbol{\mathrm{sin}}^{\mathrm{7}} \left(\boldsymbol{\mathrm{x}}\right)\right)\boldsymbol{\mathrm{dx}} \\ $$

Question Number 142910 Answers: 2 Comments: 0

Question Number 142900 Answers: 0 Comments: 3

Question Number 142893 Answers: 1 Comments: 0

.....mathematical .....analysis...... f ∈ C [0,1] and ∫_0 ^( 1) x^n f(x)dx=(1/(n+2)) , n∈N prove f(x):=x .....

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:.....{analysis}...... \\ $$$$\:\:\:\:\:\:\:{f}\:\in\:{C}\:\left[\mathrm{0},\mathrm{1}\right]\:{and}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{{n}+\mathrm{2}}\:,\:{n}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:{f}\left({x}\right):={x}\:..... \\ $$

Question Number 142875 Answers: 1 Comments: 0

Prove that ζ(s)=Π_(prime) (1/(1−p^(−s) ))

$${Prove}\:{that}\:\zeta\left({s}\right)=\underset{{prime}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−{p}^{−{s}} } \\ $$

Question Number 142805 Answers: 0 Comments: 0

Question Number 142791 Answers: 1 Comments: 0

Evaluate:: ... Ω :=∫_0 ^( 1) ((li_2 ((√x) ))/(1+(√x))) dx=?? ...........

$$\:\:\:{Evaluate}::\:... \\ $$$$\:\:\:\:\:\:\:\Omega\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{li}_{\mathrm{2}} \left(\sqrt{{x}}\:\right)}{\mathrm{1}+\sqrt{{x}}}\:{dx}=?? \\ $$$$\:\:\:\:........... \\ $$

Question Number 142719 Answers: 0 Comments: 1

find the particular solution to the differential equation y^((4)) +21y^((2)) −100y=4(8−29t)e^(−2t) . solution please.

$$\mathrm{find}\:\mathrm{the}\:\mathrm{particular}\:\mathrm{solution} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\mathrm{y}^{\left(\mathrm{4}\right)} +\mathrm{21y}^{\left(\mathrm{2}\right)} −\mathrm{100y}=\mathrm{4}\left(\mathrm{8}−\mathrm{29t}\right)\mathrm{e}^{−\mathrm{2t}} . \\ $$$$\mathrm{solution}\:\mathrm{please}. \\ $$

Question Number 142763 Answers: 2 Comments: 2

Question Number 142708 Answers: 1 Comments: 0

I=∫_0 ^( α) (√(c^2 −sin^2 θ))dθ tan α=(a^2 /b^2 ) , a^2 >b^2 , c^2 >1 Perimeter of ellipse =4∫_0 ^( π/2) (√(a^2 −(a^2 −b^2 )sin^2 θ)) dθ (is that right sir?)

$$\:{I}=\int_{\mathrm{0}} ^{\:\:\alpha} \sqrt{{c}^{\mathrm{2}} −\mathrm{sin}\:^{\mathrm{2}} \theta}{d}\theta \\ $$$$\:\mathrm{tan}\:\alpha=\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:\:,\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \:\:,\:{c}^{\mathrm{2}} >\mathrm{1} \\ $$$${Perimeter}\:{of}\:{ellipse} \\ $$$$=\mathrm{4}\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \sqrt{{a}^{\mathrm{2}} −\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)\mathrm{sin}\:^{\mathrm{2}} \theta}\:{d}\theta \\ $$$$\left({is}\:{that}\:{right}\:{sir}?\right) \\ $$

Question Number 142689 Answers: 1 Comments: 0

Question Number 142687 Answers: 1 Comments: 0

∫ (dx/( (√(1−sin x)) (√(1+cos x)))) =?

$$\:\:\:\:\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:=? \\ $$

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