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IntegrationQuestion and Answers: Page 154

Question Number 108283 Answers: 1 Comments: 0

Given the function Γ defined by Γ(x)=∫_0 ^(+∞) t^(x−1) e^(−t) dt 1. What is the domain of definition of Γ ? 2. Show that ∀x∈ DΓ, xΓ(x)=Γ(x+1) and deduce the value of Γ(n), n∈N^∗ 3. Assuming ∫_0 ^(+∞) e^(−u^2 ) =((√π)/2), calculate Γ((1/2)) and deduce that Γ(n+(1/2))=(((2n)!(√π))/(2^2^n n!))

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:\Gamma\:\mathrm{defined}\:\mathrm{by}\:\Gamma\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{+\infty} \mathrm{t}^{\mathrm{x}−\mathrm{1}} \mathrm{e}^{−\mathrm{t}} \mathrm{dt} \\ $$$$\mathrm{1}.\:\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{definition}\:\mathrm{of}\:\Gamma\:? \\ $$$$\mathrm{2}.\:\:\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{x}\in\:\mathrm{D}\Gamma,\:\mathrm{x}\Gamma\left(\mathrm{x}\right)=\Gamma\left(\mathrm{x}+\mathrm{1}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\Gamma\left(\mathrm{n}\right),\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{3}.\:\:\mathrm{Assuming}\:\int_{\mathrm{0}} ^{+\infty} \mathrm{e}^{−\mathrm{u}^{\mathrm{2}} } =\frac{\sqrt{\pi}}{\mathrm{2}},\:\mathrm{calculate}\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that} \\ $$$$\:\:\:\:\:\Gamma\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\left(\mathrm{2n}\right)!\sqrt{\pi}}{\mathrm{2}^{\mathrm{2}^{\mathrm{n}} } \mathrm{n}!} \\ $$

Question Number 108282 Answers: 0 Comments: 0

Determine the nature of the integral ∫_2 ^(+∞) (√(t^2 +3t)) ln(cos((1/t))) sin^2 ((1/(ln(t))))dt

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\mathrm{2}} ^{+\infty} \sqrt{\mathrm{t}^{\mathrm{2}} +\mathrm{3t}}\:\mathrm{ln}\left(\mathrm{cos}\left(\frac{\mathrm{1}}{\mathrm{t}}\right)\right)\:\mathrm{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{t}\right)}\right)\mathrm{dt} \\ $$

Question Number 108175 Answers: 1 Comments: 3

Question Number 108169 Answers: 1 Comments: 0

((✓BeMath✓)/(≻≺)) ∫ ((x^5 −x)/(x^8 −1)) dx ?

$$\:\:\:\:\:\:\frac{\checkmark\mathcal{B}{e}\mathcal{M}{ath}\checkmark}{\succ\prec} \\ $$$$\int\:\frac{{x}^{\mathrm{5}} −{x}}{{x}^{\mathrm{8}} −\mathrm{1}}\:{dx}\:? \\ $$

Question Number 108144 Answers: 0 Comments: 0

∫ (√(ln(x)))dx

$$\int\:\sqrt{{ln}\left({x}\right)}{dx} \\ $$

Question Number 108099 Answers: 3 Comments: 1

((★BeMath⊚)/⊓) (1) ∫ x tan^(−1) (x) dx ? (2) Find the distance of the point (3,3,1) from the plane Π with equation (r^→ −i^→ −j^→ )•(i^→ −j^→ +k^→ ) = 0 , also find the point on the plane that is nearest to (3,3,1).

$$\:\:\:\:\frac{\bigstar\mathcal{B}{e}\mathcal{M}{ath}\circledcirc}{\sqcap} \\ $$$$\:\left(\mathrm{1}\right)\:\int\:{x}\:\mathrm{tan}^{−\mathrm{1}} \left({x}\right)\:{dx}\:? \\ $$$$\left(\mathrm{2}\right)\:{Find}\:{the}\:{distance}\:{of}\:{the}\:{point}\: \\ $$$$\left(\mathrm{3},\mathrm{3},\mathrm{1}\right)\:{from}\:{the}\:{plane}\:\Pi\:{with}\:{equation} \\ $$$$\left(\overset{\rightarrow} {{r}}−\overset{\rightarrow} {{i}}−\overset{\rightarrow} {{j}}\right)\bullet\left(\overset{\rightarrow} {{i}}−\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{k}}\right)\:=\:\mathrm{0}\:,\:{also}\: \\ $$$${find}\:{the}\:{point}\:{on}\:{the}\:{plane}\:{that}\:{is} \\ $$$${nearest}\:{to}\:\left(\mathrm{3},\mathrm{3},\mathrm{1}\right). \\ $$$$ \\ $$

Question Number 108095 Answers: 3 Comments: 0

((∞BeMath∞)/♠) ∫ 2x cos^(−1) (x) dx

$$\:\:\:\frac{\infty\mathcal{B}{e}\mathcal{M}{ath}\infty}{\spadesuit} \\ $$$$\:\:\:\int\:\mathrm{2}{x}\:\mathrm{cos}^{−\mathrm{1}} \left({x}\right)\:{dx}\: \\ $$

Question Number 108076 Answers: 0 Comments: 0

can someone please show how to get ∫_0 ^π sin (a sin (x)) dx=πH_0 (a) where H_0 (a) is the Struve−H−Function

$$\mathrm{can}\:\mathrm{someone}\:\mathrm{please}\:\mathrm{show}\:\mathrm{how}\:\mathrm{to}\:\mathrm{get} \\ $$$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\mathrm{sin}\:\left({a}\:\mathrm{sin}\:\left({x}\right)\right)\:{dx}=\pi\mathrm{H}_{\mathrm{0}} \:\left({a}\right) \\ $$$$\mathrm{where}\:\mathrm{H}_{\mathrm{0}} \:\left({a}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{Struve}−\mathrm{H}−\mathrm{Function} \\ $$

Question Number 108073 Answers: 3 Comments: 0

((≜BeMath≜)/≺) ∫ ((x dx)/(x^8 −1)) ?

$$\:\:\:\:\frac{\triangleq\mathcal{B}{e}\mathcal{M}{ath}\triangleq}{\prec} \\ $$$$\:\:\int\:\frac{{x}\:{dx}}{{x}^{\mathrm{8}} −\mathrm{1}}\:? \\ $$

Question Number 107965 Answers: 2 Comments: 0

((⊚BeMath⊚)/) ∫ x (√(x/(2a−x))) dx ?