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IntegrationQuestion and Answers: Page 84

Question Number 140634 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) ((cos(2sinx))/((x^2 −x+1)^2 ))dx

$f i n d \int_{- \infty}^{+ \infty} \frac{c o s (2 s i n x)}{{(x^{2} - x + 1)}^{2}} d x$

Question Number 140635 Answers: 1 Comments: 0

find ∫_(−∞) ^(+∞) ((sin(2cosx))/((x^2 −x+1)^2 ))dx

$f i n d \int_{- \infty}^{+ \infty} \frac{s i n (2 c o s x)}{{(x^{2} - x + 1)}^{2}} d x$

Question Number 140615 Answers: 1 Comments: 0

Find the area common to the curve y^2 = 12x and x^2 +y^2 = 24x .

$Find the area common$ $to the curve y^{2} = 12 x and$ $x^{2} + y^{2} = 24 x .$

Question Number 140614 Answers: 2 Comments: 0

∫ _0^(π/2) ln (sin x) sec^2 x dx =?

$\int_{0}^{\frac{π}{2}} \ln (\sin x) \sec^{2} x dx = ?$

Question Number 140588 Answers: 1 Comments: 0

.......Advanced ....★★★....Calculus....... evaluation the value of : 𝛗 :=∫_0 ^( (π/2)) sin^2 (x).ln(sin(x))dx solution:: ξ (a):=∫_0 ^( (π/2)) sin^(2+a) (x)dx =(1/2)β (((3+a)/2) ,(1/2)) :=(1/2)(((Γ(((3+a)/2))Γ((1/2)))/(Γ(2+(a/2))))).......✓ 𝛗:= ξ ′ (0) ..............✓ :=(1/2) (√π) (( Γ′(((3+a)/2)).Γ(2+(a/2))−Γ(((3+a)/2)).Γ′(2+(a/2)))/(Γ^2 (2+(a/2)))) ∣_(a=0) :=(1/2)(√π) ((Γ′((3/2))−Γ((3/2)).Γ′(2))/(( Γ^2 (2):=1 ))) :=(1/2)(√π) ((ψ((3/2))Γ((3/2))−Γ((3/2)).ψ(2))/1) := ((√π)/4){ (2−γ−2ln(2)−(1−γ)} :=((√π)/4)(1−ln(4))=(√π) ln(((e/4))^(1/4) )

$. . . . . . . A d v a n c e d . . . . ★ ★ ★ . . . . C a l c u l u s . . . . . . .$ $e v a l u a t i o n t h e v a l u e o f :$ $ϕ := \int_{0}^{\frac{π}{2}} {s i n}^{2} (x) . l n (s i n (x)) d x$ $s o l u t i o n ::$ $ξ (a) := \int_{0}^{\frac{π}{2}} {s i n}^{2 + a} (x) d x = \frac{1}{2} β (\frac{3 + a}{2}, \frac{1}{2})$ $:= \frac{1}{2} (\frac{Γ (\frac{3 + a}{2}) Γ (\frac{1}{2})}{Γ (2 + \frac{a}{2})}) . . . . . . . ✓$ $ϕ := ξ^{'} (0) . . . . . . . . . . . . . . ✓$ $:= \frac{1}{2} \sqrt{π} \frac{Γ^{'} (\frac{3 + a}{2}) . Γ (2 + \frac{a}{2}) - Γ (\frac{3 + a}{2}) . Γ^{'} (2 + \frac{a}{2})}{Γ^{2} (2 + \frac{a}{2})} ∣_{a = 0}$ $:= \frac{1}{2} \sqrt{π} \frac{Γ^{'} (\frac{3}{2}) - Γ (\frac{3}{2}) . Γ^{'} (2)}{(Γ^{2} (2) := 1)}$ $:= \frac{1}{2} \sqrt{π} \frac{ψ (\frac{3}{2}) Γ (\frac{3}{2}) - Γ (\frac{3}{2}) . ψ (2)}{1}$ $:= \frac{\sqrt{π}}{4} {(2 - γ - 2 l n (2) - (1 - γ)}$ $:= \frac{\sqrt{π}}{4} (1 - l n (4)) = \sqrt{π} l n (\sqrt[4]{\frac{e}{4}})$

Question Number 140554 Answers: 0 Comments: 5

What′s the relationship between Dirichlet β(s) function with ζ(s) function ? That is Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^s )) with Σ_(n=1) ^∞ (1/n^s ).

${W h a t}^{'} s t h e r e l a t i o n s h i p b e t w e e n D i r i c h l e t β (s) f u n c t i o n w i t h$ $ζ (s) f u n c t i o n ? T h a t i s \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(2 n + 1)}^{s}} w i t h \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{s}} .$

Question Number 140500 Answers: 0 Comments: 0

tan^2 1°+tan^2 2°+tan^2 3°+...+tan^2 89°=((15931)/3) ???

$\tan^{2} 1 ° + \tan^{2} 2 ° + \tan^{2} 3 ° + . . . + \tan^{2} 89 ° = \frac{15931}{3} ? ? ?$

Question Number 140490 Answers: 1 Comments: 1

Question Number 140447 Answers: 1 Comments: 0

Question Number 140405 Answers: 1 Comments: 0

find the value of :: Θ :=Σ_(n=1 ) ^∞ (1/(4n.(4n+1).(4n+2).(4n+3)))=?

$f i n d t h e v a l u e o f ::$ $Θ := \underset{n = 1}{\sum^{\infty}} \frac{1}{4 n . (4 n + 1) . (4 n + 2) . (4 n + 3)} = ?$

Question Number 140401 Answers: 2 Comments: 0

evaluate :: Φ:=∫_0 ^( ∞) xe^(−(x^2 /4)) ln(x)dx = m.( π γ) find ” m ” ......

$e v a l u a t e ::$ $Φ := \int_{0}^{\infty} {x e}^{- \frac{x^{2}}{4}} l n (x) d x = m . (π γ)$ $f i n d^{″} m^{″} . . . . . .$

Question Number 140399 Answers: 1 Comments: 0

𝛏 :=∫_0 ^( ∞) ((e^(−x^2 ) −e^(−x) )/x) dx = k.γ find ” k ” ... γ := Euler constant....

$ξ := \int_{0}^{\infty} \frac{e^{- x^{2}} - e^{- x}}{x} d x = k . γ$ $f i n d^{″} k^{″} . . .$ $γ := E u l e r c o n s t a n t . . . .$

Question Number 140388 Answers: 3 Comments: 0

∫_0 ^∞ ((ln x)/((x^2 +a^2 )^5 )) dx

$\int_{0}^{\infty} \frac{\ln x}{{(x^{2} + a^{2})}^{5}} dx$

Question Number 140353 Answers: 0 Comments: 1

Question Number 140350 Answers: 0 Comments: 0

Question Number 140334 Answers: 1 Comments: 0

calculate :: 𝛏 := ∫_(−∞) ^( ∞) ln(2−2cos(x^2 ))dx=?

$c a l c u l a t e ::$ $ξ := \int_{- \infty}^{\infty} l n (2 - 2 c o s (x^{2})) d x = ?$

Question Number 140330 Answers: 2 Comments: 0

Find the Integration Value: 1 ∫(((√x)d(x))/(1+^3 (√x)))=? 2 ∫(dx/(x^(1/2) −x^(1/4) ))=?

$F i n d t h e I n t e g r a t i o n V a l u e :$ $1 \int \frac{\sqrt{x} d (x)}{1 +^{3} \sqrt{x}} = ?$ $2 \int \frac{d x}{x^{\frac{1}{2}} - x^{\frac{1}{4}}} = ?$

Question Number 140310 Answers: 2 Comments: 0

prove that ∫_0 ^∞ ((ln x)/(x^2 +1)) dx = 0

$prove that \int_{0}^{\infty} \frac{\ln x}{x^{2} + 1} dx = 0$

Question Number 140282 Answers: 0 Comments: 0

Σ_(k=0) ^(p−1) ((p),(k) )sin [2(p−k)x]=? ((p),(0) )sin (2px)+ ((p),(1) )sin [(2p−2)x]+ ((p),(2) )sin [(2p−4)x]+...+ ((( p)),((p−1)) )sin (2x)=2^p ∙cos^p (x)∙sin (px) ??? or ∫_0 ^∞ ((cos^p (x)∙sin (px))/x)dx=(π/2)(1−2^(−p) ) why ???

$\underset{k = 0}{\sum^{p - 1}} (\begin{matrix} p \\ k \end{matrix}) \sin [2 (p - k) x] = ?$ $(\begin{matrix} p \\ 0 \end{matrix}) \sin (2 p x) + (\begin{matrix} p \\ 1 \end{matrix}) \sin [(2 p - 2) x] + (\begin{matrix} p \\ 2 \end{matrix}) \sin [(2 p - 4) x] + . . . + (\begin{matrix} p \\ p - 1 \end{matrix}) \sin (2 x) = 2^{p} \cdot \cos^{p} (x) \cdot \sin (p x) ? ? ?$ $o r \int_{0}^{\infty} \frac{\cos^{p} (x) \cdot \sin (p x)}{x} d x = \frac{π}{2} (1 - 2^{- p}) w h y ? ? ?$