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IntegrationQuestion and Answers: Page 129

Question Number 119852 Answers: 1 Comments: 0

lim_(n→∞) n^2 ∫ _0 ^(1/n) x^(x+1) dx =?

$\lim_{n \to \infty} n^{2} \int \underset{0}{\overset{\frac{1}{n}}{}} x^{x + 1} d x = ?$

Question Number 119784 Answers: 2 Comments: 0

∫ (dx/( (√((4x−x^2 )^3 ))))

$\int \frac{d x}{\sqrt{{(4 x - x^{2})}^{3}}}$

Question Number 119773 Answers: 3 Comments: 0

∫_(−4) ^4 x^3 (√(16−x^2 )) sec x dx

$\underset{- 4}{\int^{4}} x^{3} \sqrt{16 - x^{2}} \sec x d x$

Question Number 119821 Answers: 3 Comments: 0

∫_(−3) ^0 ((6x−6)/( (√(x^2 −2x+1)))) dx =?

$\underset{- 3}{\int^{0}} \frac{6 x - 6}{\sqrt{x^{2} - 2 x + 1}} d x = ?$

Question Number 119762 Answers: 3 Comments: 0

calculate ∫_0 ^∞ ((x^4 dx)/((2x+1)^5 (3x+1)^8 ))

$calculate \int_{0}^{\infty} \frac{x^{4} dx}{{(2 x + 1)}^{5} {(3 x + 1)}^{8}}$

Question Number 119752 Answers: 1 Comments: 0

find I_λ =∫_0 ^∞ ((ch(1+λcosx))/((x^2 +1)^2 ))dx (λ real >0)

$f i n d I_{λ} = \int_{0}^{\infty} \frac{c h (1 + λ c o s x)}{{(x^{2} + 1)}^{2}} d x$ $(λ r e a l > 0)$

Question Number 119750 Answers: 1 Comments: 0

Π_(k=1) ^(1019) [((2k)/(2k−1))]=?

$\underset{k = 1}{\prod^{1019}} [\frac{2 k}{2 k - 1}] = ?$

Question Number 119696 Answers: 2 Comments: 0

For a<b then ∫_a ^b (x−a)(x−b) dx equal to _

$F o r a < b t h e n \underset{a}{\int^{b}} (x - a) (x - b) d x$ $e q u a l t o_$

Question Number 119681 Answers: 4 Comments: 0

∫_0 ^π (√((1+cos2x)/2)) dx ∫_0 ^∞ [ne^(−x) ]dx

$\int_{0}^{π} \sqrt{\frac{1 + c o s 2 x}{2}} d x$ $\int_{0}^{\infty} [{n e}^{- x}] d x$

Question Number 119570 Answers: 3 Comments: 0

... ♣_♣ ^♣ nice calculus♣_♣ ^♣ ... prove that :: Ω=∫_0 ^( ∞) e^(−2x) ln(((1+e^(−x) )/(1−e^(−x) )))=1 ...★ M.N.1970★...

$. . . \underset{♣}{\overset{♣}{♣}} n i c e c a l c u l u s \underset{♣}{\overset{♣}{♣}} . . .$ $p r o v e t h a t ::$ $Ω = \int_{0}^{\infty} e^{- 2 x} l n (\frac{1 + e^{- x}}{1 - e^{- x}}) = 1$ $. . . ★ M . N . 1970 ★ . . .$

Question Number 119591 Answers: 0 Comments: 6

...nice calculus... prove that :: ∫_0 ^( (π/2)) (√(((2^x −1)sin^3 (x))/((2^x +1)(sin^3 (x)+cos^3 (x))))) dx<(π/8) ...m.n.1970...

$. . . n i c e c a l c u l u s . . .$ $p r o v e t h a t ::$ $\int_{0}^{\frac{π}{2}} \sqrt{\frac{(2^{x} - 1) {s i n}^{3} (x)}{(2^{x} + 1) ({s i n}^{3} (x) + {c o s}^{3} (x))}} d x < \frac{π}{8}$ $. . . m . n . 1970 . . .$

Question Number 119462 Answers: 2 Comments: 0

... advanced calculus... evaluate :: Ω=∫_0 ^( ∞) ((tan^(−1) (x))/(e^(2πx) −1))dx =? m.n.1970

$. . . a d v a n c e d c a l c u l u s . . .$ $e v a l u a t e ::$ $Ω = \int_{0}^{\infty} \frac{{t a n}^{- 1} (x)}{e^{2 π x} - 1} d x = ?$ $m . n . 1970$

Question Number 119446 Answers: 0 Comments: 0

calculste ∫_0 ^∞ ((ln(2+x^2 ))/(1+x^3 ))dx

$c a l c u l s t e \int_{0}^{\infty} \frac{l n (2 + x^{2})}{1 + x^{3}} d x$

Question Number 119445 Answers: 0 Comments: 0

...nice calculus... prove that:: Σ_(n=1) ^∞ (((−1)^(n−1) )/(n^3 (((2n)),(n) ))) =^(???) ζ(3) m.n.1970

$. . . n i c e c a l c u l u s . . .$ $p r o v e t h a t ::$ $\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{3} (\begin{matrix} 2 n \\ n \end{matrix})} \overset{? ? ?}{=} ζ (3)$ $m . n . 1970$

Question Number 119442 Answers: 0 Comments: 0

... advanced calculus... prove that : Σ_(n=1) ^∞ (1/(n^2 (((2n)),(n) ))) =^(???) ((ζ(2))/3) solution:: Σ_(n=1) ^∞ (1/(n^2 ∗(((2n)!)/((n!)^2 ))))=Σ_(n=1) ^∞ ((n!∗n!)/(n^2 ∗(2n)!)) =Σ_(n=1) ^∞ ((Γ(n)Γ(n+1))/(nΓ(2n+1)))=Σ_(n=1) ^∞ ((β(n,n+1))/n) =Σ_(n=1) ^∞ (1/n)∫_0 ^( 1) x^(n−1) (1−x)^n =∫_0 ^( 1) (1/x)Σ(((x−x^2 )^n )/n)dx =−∫_0 ^( 1) ((ln(1−x+x^2 ))/x)dx =−∫_0 ^( 1) ((ln(1+x^3 )−ln(1+x))/x)dx =∫_0 ^( 1) ((ln(1+x))/x)dx −∫_0 ^( 1) ((ln(1+x^3 ))/x)dx =−li_2 (−1) −∫_0 ^( 1) ((Σ_(n=1) (((−1)^(n+1) x^(3n) )/n))/x) dx =(π^2 /(12))+Σ_(n=1) ^∞ (((−1)^n )/n)∫_0 ^( 1) x^(3n−1) dx =(π^2 /(12)) +(1/3)Σ_(n=1) ^∞ (((−1)^n )/n^2 ) =(π^2 /(12))−(π^2 /(36)) =(π^2 /(18)) =((ζ(2))/3) ✓✓ m.n.july.1970..

$. . . a d v a n c e d c a l c u l u s . . .$ $p r o v e t h a t : \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2} (\begin{matrix} 2 n \\ n \end{matrix})} \overset{? ? ?}{=} \frac{ζ (2)}{3}$ $s o l u t i o n :: \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2} * \frac{(2 n)!}{{(n!)}^{2}}} = \underset{n = 1}{\sum^{\infty}} \frac{n! * n!}{n^{2} * (2 n)!}$ $= \underset{n = 1}{\sum^{\infty}} \frac{Γ (n) Γ (n + 1)}{n Γ (2 n + 1)} = \underset{n = 1}{\sum^{\infty}} \frac{β (n, n + 1)}{n}$ $= \underset{n = 1}{\sum^{\infty}} \frac{1}{n} \int_{0}^{1} x^{n - 1} {(1 - x)}^{n}$ $= \int_{0}^{1} \frac{1}{x} Σ \frac{{(x - x^{2})}^{n}}{n} d x$ $= - \int_{0}^{1} \frac{l n (1 - x + x^{2})}{x} d x$ $= - \int_{0}^{1} \frac{l n (1 + x^{3}) - l n (1 + x)}{x} d x$ $= \int_{0}^{1} \frac{l n (1 + x)}{x} d x - \int_{0}^{1} \frac{l n (1 + x^{3})}{x} d x$ $= - {l i}_{2} (- 1) - \int_{0}^{1} \frac{\sum_{n = 1} \frac{{(- 1)}^{n + 1} x^{3 n}}{n}}{x} d x$ $= \frac{π^{2}}{12} + \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n} \int_{0}^{1} x^{3 n - 1} d x$ $= \frac{π^{2}}{12} + \frac{1}{3} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n^{2}} = \frac{π^{2}}{12} - \frac{π^{2}}{36}$ $= \frac{π^{2}}{18} = \frac{ζ (2)}{3} ✓ ✓$ $m . n . j u l y . 1970 . .$

Question Number 119425 Answers: 1 Comments: 0

decompose F(x) =((2x−1)/((x^2 −1)^2 (x^2 +3))) and calculate ∫_(√2) ^(+∞) F(x)dx

$decompose F (x) = \frac{2 x - 1}{{(x^{2} - 1)}^{2} (x^{2} + 3)}$ $and calculate \int_{\sqrt{2}}^{+ \infty} F (x) dx$

Question Number 119335 Answers: 2 Comments: 0

Express f(x) = (1/((x−1)^2 (x^2 +1))) into partial fractions. hence evaluate I = ∫_0 ^4 f(x) dx

$Express f (x) = \frac{1}{{(x - 1)}^{2} (x^{2} + 1)} into partial fractions .$ $hence evaluate I = \underset{0}{\int^{4}} f (x) d x$

Question Number 119323 Answers: 1 Comments: 4

Question Number 119306 Answers: 3 Comments: 0

Question Number 119282 Answers: 1 Comments: 0

... nice calculus... evaluate:: I:= ∫_0 ^( 1) li_2 (1−x^2 )dx=?? .m.n.1970.

$. . . n i c e c a l c u l u s . . .$ $e v a l u a t e ::$ $I := \int_{0}^{1} {l i}_{2} (1 - x^{2}) d x = ? ?$ $. m . n . 1970 .$

Question Number 119318 Answers: 0 Comments: 0

find ∫_0 ^∞ ((lnx)/(x^4 +x^2 +2))dx

$f i n d \int_{0}^{\infty} \frac{l n x}{x^{4} + x^{2} + 2} d x$

Question Number 119317 Answers: 0 Comments: 0

calculate ∫_0 ^(2π) (dθ/((x^2 −2cosθ x+1)^2 ))

$c a l c u l a t e \int_{0}^{2 π} \frac{d θ}{{(x^{2} - 2 c o s θ x + 1)}^{2}}$

Question Number 119262 Answers: 3 Comments: 0

∫ (x^2 /( (√((4−x^2 )^5 )))) dx

$\int \frac{x^{2}}{\sqrt{{(4 - x^{2})}^{5}}} d x$

Question Number 119256 Answers: 0 Comments: 0

Determine ∫_(−(π/4)) ^( (π/4)) (cost+(√(1+t^2 sin^3 tcos^3 t))dt=?

$D e t e r m i n e \int_{- \frac{π}{4}}^{\frac{π}{4}} (c o s t + \sqrt{1 + t^{2} {s i n}^{3} {t c o s}^{3} t} d t = ?$

Question Number 119151 Answers: 0 Comments: 0

Question Number 119070 Answers: 2 Comments: 0

∫ ((x^2 −x+6)/(x^3 +3x)) dx ∫ ((5x^2 +3x−2)/(x^3 +2x^2 )) dx

$\int \frac{x^{2} - x + 6}{x^{3} + 3 x} d x$ $\int \frac{5 x^{2} + 3 x - 2}{x^{3} + 2 x^{2}} d x$

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