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IntegrationQuestion and Answers: Page 107

Question Number 128922 Answers: 1 Comments: 0

...advanced calculus... evaluate ::: Ω=∫_(0 ) ^( (1/2)) ((Arctanh(x))/x)dx=?

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

Question Number 128900 Answers: 3 Comments: 0

∫ ((4x+5)/((x+2)(x+3)(x+4)(x+5)+1)) dx?

$\int \frac{4 x + 5}{(x + 2) (x + 3) (x + 4) (x + 5) + 1} dx ?$

Question Number 128888 Answers: 1 Comments: 0

∫ (((x−1)(x−2)(x−3))/((x−4)(x−5)(x−6))) dx =?

$\int \frac{(x - 1) (x - 2) (x - 3)}{(x - 4) (x - 5) (x - 6)} dx = ?$

Question Number 128853 Answers: 1 Comments: 0

F(x)=∫_x ^(2x) (dx/( (√(t^4 +t^2 +1)))) 1. Show that F is defined, continuous and derivable in R

$F (x) = \int_{x}^{2 x} \frac{dx}{\sqrt{t^{4} + t^{2} + 1}}$ $1 . Show that F is defined, continuous and derivable in R$

Question Number 128851 Answers: 3 Comments: 2

...nice calculus (I)... calculate :: Ψ=∫_0 ^( 1) {(e−1)(√(log( 1+ex−x ))) +e^x^2 }dx=?

$. . . n i c e c a l c u l u s (I) . . .$ $c a l c u l a t e ::$ $Ψ = \int_{0}^{1} {(e - 1) \sqrt{l o g (1 + e x - x)} + e^{x^{2}}} d x = ?$

Question Number 128841 Answers: 1 Comments: 2

...nice calculus ... Evaluation of :: Φ= ∫_0 ^( 1) ln(x).arctan(x)dx solution:: note 1:: Σ_(n=1) ^∞ (((−1)^(n−1) )/(2n−1))=arctan(1)=(π/4) note 2 :: Σ_(n=1) ^∞ (((−1)^(n−1) )/n) =ln(1+1)=ln(2) note 3:: Σ_(n=1) ^∞ (((−1)^(n−1) )/n^2 ) =_(eta function) ^(Drichlet) η(2)=(π^2 /(12)) ......start....... Φ=∫_0 ^( 1) {ln(x)Σ_(n=1 ) ^∞ (((−1)^(n−1) )/(2n−1)))x^(2n−1) }dx =Σ_(n=1) ^∞ (((−1)^(n−1) )/((2n−1)))∫_0 ^( 1) ln(x)x^(2n−1) dx =Σ_(n=1 ) ^∞ (((−1)^(n−1) )/((2n−1))){[(x^(2n) /(2n))ln(x)]_0 ^1 −(1/(2n))∫_0 ^( 1) x^(2n−1) dx} =(1/4)Σ_(n=1) ^∞ (((−1)^n )/((2n−1)n^2 ))=(1/4)Σ_(n=1) ^∞ (−1)^n [((2n−(2n−1))/((2n−1)n^2 ))] =(1/2)Σ_(n=1) ^∞ (((−1)^n )/((2n−1)n)) −(1/4)Σ_(n=1) ^∞ (((−1)^n )/n^2 ) =(1/2)Σ_(n=1) ^∞ ((2(−1)^n )/(2n−1))−(1/2)Σ_(n=1) ^∞ (((−1)^n )/n)+(1/4)η(2) =((−π)/4)+(1/2)ln(2)+(π^2 /(48)) ... ...Φ=(π^2 /(48)) −(π/4) +(1/2)ln(2) ... ...m.n.july.1970...

Question Number 128826 Answers: 1 Comments: 0

∫_(−1) ^( 5) (√((2x^2 −8)/x)) dx =?

$\int_{- 1}^{5} \sqrt{\frac{{2 x}^{2} - 8}{x}} dx = ?$

Question Number 128797 Answers: 2 Comments: 0

...nice calculus... φ =^(???) ∫_0 ^( ∞) (((tanh(x))/e^x )) dx

$. . . n i c e c a l c u l u s . . .$ $ϕ \overset{? ? ?}{=} \int_{0}^{\infty} (\frac{t a n h (x)}{e^{x}}) d x$

Question Number 128775 Answers: 3 Comments: 0

∫ (dx/((1−x)^2 (√(1−x^2 )))) ?

$\int \frac{dx}{{(1 - x)}^{2} \sqrt{1 - x^{2}}} ?$

Question Number 128750 Answers: 1 Comments: 1

Given a function f satisfy f(−x)=3f(x). If ∫_(−1) ^( 2) f(x) dx = 2 then ∫_(−2) ^( 1) f(x)dx=?

$Given a function f satisfy f (- x) = 3 f (x) .$ $If \int_{- 1}^{2} f (x) dx = 2 then \int_{- 2}^{1} f (x) dx = ?$

Question Number 128736 Answers: 1 Comments: 1

... nice calculus... evluate :: φ = ∫_0 ^( ∞) e^(−x^2 ) cos(x)dx=?

$. . . n i c e c a l c u l u s . . .$ $e v l u a t e ::$ $ϕ = \int_{0}^{\infty} e^{- x^{2}} c o s (x) d x = ?$

Question Number 128721 Answers: 0 Comments: 1

∫_0 ^1 ((ln x)/(x(x^2 +1))) dx

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

Question Number 128710 Answers: 0 Comments: 0

∫e^x (((1+sinx+cosx)/(cos^2 x))) dx

$\int e^{x} (\frac{1 + s i n x + c o s x}{{c o s}^{2} x}) d x$

Question Number 128707 Answers: 1 Comments: 0

...nice calculus... prove that:: ∫_0 ^( ∞) ((ln(1+ϕ^2 x^2 ))/(1+π^2 x^2 )) dx=ln(((π+ϕ)/π)) ϕ::= golen ratio...

$. . . nice calculus . . .$ $p r o v e t h a t ::$ $\int_{0}^{\infty} \frac{l n (1 + φ^{2} x^{2})}{1 + π^{2} x^{2}} d x = l n (\frac{π + φ}{π})$ $φ ::= g o l e n r a t i o . . .$

Question Number 128702 Answers: 1 Comments: 0

If ((sin^4 x)/2) + ((cos^4 x)/3) = (1/5) then ((sin^8 x)/8) + ((cos^8 x)/(27)) = ?

$If \frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5} then$ $\frac{\sin^{8} x}{8} + \frac{\cos^{8} x}{27} = ?$

Question Number 128680 Answers: 1 Comments: 0

...nice calculus... Σ_(n=0) ^∞ (1/((3n+1)ϕ^(3n+1) )) =? ϕ :: golden ratio...

$. . . n i c e c a l c u l u s . . .$ $\underset{n = 0}{\sum^{\infty}} \frac{1}{(3 n + 1) φ^{3 n + 1}} = ?$ $φ :: g o l d e n r a t i o . . .$

Question Number 128664 Answers: 2 Comments: 0

∫_0 ^( π/2) (1−sin x+sin^2 x−sin^3 x+sin^4 x−sin^5 x+...) dx =?

$\int_{0}^{π / 2} (1 - \sin x + \sin^{2} x - \sin^{3} x + \sin^{4} x - \sin^{5} x + . . .) dx = ?$

Question Number 128634 Answers: 1 Comments: 0

θ = ∫ (1+4x^4 )e^x^4 dx

$θ = \int (1 + {4 x}^{4}) e^{x^{4}} dx$

Question Number 128633 Answers: 2 Comments: 0

Ω = ∫_0 ^( (1/3)) x^(2n) ln(1−x)dx

$Ω = \int_{0}^{\frac{1}{3}} x^{2 n} \ln (1 - x) dx$

Question Number 128620 Answers: 1 Comments: 0

Question Number 128610 Answers: 1 Comments: 1

∫_(−π/4) ^( π/4) ((sec x)/(e^x +1)) dx

$\int_{- π / 4}^{π / 4} \frac{\sec x}{e^{x} + 1} dx$

Question Number 128608 Answers: 1 Comments: 0

∫ x^2 .tan^(−1) ((x/2))dx=?

$\int x^{2} . \tan^{- 1} (\frac{x}{2}) dx = ?$

Question Number 128602 Answers: 1 Comments: 0

∫_(−1) ^1 ∫_0 ^(1−x) (√((x^(2/3) y−x^(5/3) y−x^(2/3) y^2 )/y^2 ))dydx

$\int_{- 1}^{1} \int_{0}^{1 - x} \sqrt{\frac{x^{\frac{2}{3}} y - x^{\frac{5}{3}} y - x^{\frac{2}{3}} y^{2}}{y^{2}}} d y d x$

Question Number 128575 Answers: 3 Comments: 3

∫(√(x^2 +4x+13))dx=??

$\int \sqrt{x^{2} + 4 x + 13} d x = ? ?$

Question Number 128570 Answers: 0 Comments: 0

... mathematical analysis... if ′′ f ′′ is Reimann integrable function on [a , b ] , then prove:: lim_(t→∞ ) {∫_a ^( b) f(x)cos(tx)dx }=0 ..Reimann−Lebesgue theorem...

$. . . m a t h e m a t i c a l a n a l y s i s . . .$ $i f^{″} f^{″} i s R e i m a n n i n t e g r a b l e$ $f u n c t i o n o n [a, b], t h e n p r o v e ::$ ${l i m}_{t \to \infty} {\int_{a}^{b} f (x) c o s (t x) d x} = 0$ $. . R e i m a n n - L e b e s g u e t h e o r e m . . .$

Question Number 128542 Answers: 1 Comments: 0

∫ (((x^4 −x)^(1/4) )/x^5 ) dx =?

$\int \frac{{(x^{4} - x)}^{1 / 4}}{x^{5}} dx = ?$

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