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IntegrationQuestion and Answers: Page 67

Question Number 151341 Answers: 1 Comments: 0

∫_α ^β (dx/(x(√((x−α)(β−x) ))))=(π/( (√(αβ)))) where α,β >0

$\int_{α}^{β} \frac{dx}{x \sqrt{(x - α) (β - x)}} = \frac{π}{\sqrt{α β}}$ $where α, β > 0$

Question Number 151340 Answers: 1 Comments: 0

Evaluate ∫_(−1) ^1 (((2x^(332) +x^(998) +4x^(1668) .sin x^(691) ))/(1+x^(666) ))dx

$Evaluate$ $\int_{- 1}^{1} \frac{({2 x}^{332} + x^{998} + {4 x}^{1668} . \sin x^{691})}{1 + x^{666}} dx$

Question Number 151287 Answers: 0 Comments: 2

I = ∫_(x=a) ^( x=b) (√(u^2 + v^2 x^2 − 2uvwx)) dx = ?

$I = \int_{x = a}^{x = b} \sqrt{u^{2} + v^{2} x^{2} - 2 u v w x} d x = ?$

Question Number 151272 Answers: 0 Comments: 1

Question Number 151268 Answers: 0 Comments: 0

Σ_(r=1) ^∞ (((−1)^(r−1) )/r)[ψ((r/2)+(1/4))−ψ((r/2)−(1/4))]

$\underset{r = 1}{\sum^{\infty}} \frac{{(- 1)}^{r - 1}}{r} [ψ (\frac{r}{2} + \frac{1}{4}) - ψ (\frac{r}{2} - \frac{1}{4})]$

Question Number 151246 Answers: 1 Comments: 0

find I=∫_0 ^(π/4) ln(cosx)dx and J=∫_0 ^(π/4) ln(sinx)dx

$find I = \int_{0}^{\frac{π}{4}} \ln (cosx) dx and J = \int_{0}^{\frac{π}{4}} \ln (sinx) dx$

Question Number 151230 Answers: 4 Comments: 2

prove: ∫_0 ^( ∞) (( ln ( 1+x^( 2) ))/(x^( 2) (1+x^( 2) )))dx= π ln((e/2) ) ..

$p r o v e :$ $\int_{0}^{\infty} \frac{l n (1 + x^{2})}{x^{2} (1 + x^{2})} d x = π l n (\frac{e}{2}) . .$

Question Number 151221 Answers: 1 Comments: 2

∫((sin x)/(sin (x−a)))dx

$\int \frac{\sin x}{\sin (x - a)} dx$

Question Number 151220 Answers: 3 Comments: 0

show that ∫_0 ^(π/2) ((cos x)/(cos x+sin x+1))dx=(1/4)(π−2ln 2)

$show that$ $\int_{0}^{\frac{π}{2}} \frac{\cos x}{\cos x + \sin x + 1} dx = \frac{1}{4} (π - 2 \ln 2)$

Question Number 151182 Answers: 0 Comments: 0

∫_0 ^( ∞) ((sin(2x)ln(x))/x) dx= m.∫_0 ^( ∞) (( ln(1+2x+x^2 ))/(x(ln^2 (x)+ π^( 2) ))) dx m=?....

$\int_{0}^{\infty} \frac{s i n (2 x) l n (x)}{x} d x = m . \int_{0}^{\infty} \frac{l n (1 + 2 x + x^{2})}{x ({l n}^{2} (x) + π^{2})} d x$ $m = ? . . . .$

Question Number 151117 Answers: 0 Comments: 0

Question Number 151069 Answers: 0 Comments: 0

Calculate :: ∫_0 ^∞ ((sin (1145141919810893x))/(x(cosh x+cos x)))dx=(π/4)

$Calculate :: \int_{0}^{\infty} \frac{\sin (1145141919810893 x)}{x (\cosh x + \cos x)} dx = \frac{π}{4}$

Question Number 151068 Answers: 0 Comments: 0

Calculate :: ∫_0 ^∞ ((√x)/((x^4 +14x^2 +1)^(5/4) ))dx=((Γ^2 ((3/4)))/(4(√(2π))))

$Calculate :: \int_{0}^{\infty} \frac{\sqrt{x}}{{(x^{4} + {14 x}^{2} + 1)}^{\frac{5}{4}}} dx = \frac{Γ^{2} (\frac{3}{4})}{4 \sqrt{2 π}}$

Question Number 151067 Answers: 0 Comments: 0

Calculate :: ∫_0 ^π sin (x/2)∙arctan((2/(sin x))−1)dx=(√2)ln(1+(√2))+(1−(1/( (√2))))π

$Calculate :: \int_{0}^{π} \sin \frac{x}{2} \cdot \arctan (\frac{2}{\sin x} - 1) dx = \sqrt{2} \ln (1 + \sqrt{2}) + (1 - \frac{1}{\sqrt{2}}) π$

Question Number 151066 Answers: 0 Comments: 0

Calculate :: ∫_0 ^(π/2) (arctan(((sin x)/2))+arctan(((cos 3x+15cos x)/8)))dx =(π^2 /4)−ln^2 (1+(√2))

$Calculate :: \int_{0}^{π / 2} (\arctan (\frac{\sin x}{2}) + \arctan (\frac{\cos 3 x + 15 \cos x}{8})) dx$ $= \frac{π^{2}}{4} - \ln^{2} (1 + \sqrt{2})$

Question Number 151065 Answers: 0 Comments: 0

Calculate :: ∫_(−∞) ^(+∞) ((Γ(x))/(Γ(x+a)))sin (πx)dx=(2^(a−1) /(Γ(a)))π (a>0)

$Calculate :: \int_{- \infty}^{+ \infty} \frac{Γ (x)}{Γ (x + a)} \sin (π x) dx = \frac{2^{a - 1}}{Γ (a)} π (a > 0)$

Question Number 151064 Answers: 0 Comments: 0

Calculate :: ∫_0 ^π arctan(((2sin^2 x)/(1−2(√2)ϕcos x+2ϕ^2 )))dx=πarctan (√ϕ) (ϕ=(((√5)−1)/2))

$Calculate :: \int_{0}^{π} \arctan (\frac{2 \sin^{2} x}{1 - 2 \sqrt{2} φ \cos x + 2 φ^{2}}) dx = π \arctan \sqrt{φ} (φ = \frac{\sqrt{5} - 1}{2})$

Question Number 151063 Answers: 0 Comments: 0

Calculate :: ∫_0 ^∞ ((ln(1+x))/((π^2 +ln^2 x)x))dx=γ

$Calculate :: \int_{0}^{\infty} \frac{\ln (1 + x)}{(π^{2} + \ln^{2} x) x} dx = γ$

Question Number 151062 Answers: 0 Comments: 0

Calculate :: ∫_0 ^∞ (dx/( (√(x^2 +x))∙((8x^2 +8x+1))^(1/4) ))=((Γ^2 ((1/8)))/(2^((11)/4) Γ((1/4))))

$Calculate :: \int_{0}^{\infty} \frac{dx}{\sqrt{x^{2} + x} \cdot \sqrt[4]{{8 x}^{2} + 8 x + 1}} = \frac{Γ^{2} (\frac{1}{8})}{2^{\frac{11}{4}} Γ (\frac{1}{4})}$

Question Number 151061 Answers: 0 Comments: 0

Calculate :: ∫_0 ^(π/2) x∙cot x∙ln^2 cos xdx=(π^3 /(24))ln2+(π/6)ln^3 2−(3/(16))πζ(3)

$Calculate :: \int_{0}^{π / 2} x \cdot \cot x \cdot \ln^{2} \cos xdx = \frac{π^{3}}{24} \ln 2 + \frac{π}{6} \ln^{3} 2 - \frac{3}{16} π ζ (3)$

Question Number 151060 Answers: 0 Comments: 0

Calculate :: ∫_0 ^∞ ((x(√x))/((x^2 +1)(1+ax)))dx=((a^2 −a+(√(2a)))/( (√2)a(1+a^2 )))π ,(a>0)

$Calculate :: \int_{0}^{\infty} \frac{x \sqrt{x}}{(x^{2} + 1) (1 + ax)} dx = \frac{a^{2} - a + \sqrt{2 a}}{\sqrt{2} a (1 + a^{2})} π, (a > 0)$

Question Number 151037 Answers: 1 Comments: 0

Σ_(r=1,r≠s) ^n Σ_(s=1) ^n ((rs)/(n(n−1)))=^? (((n+1)(3n+2))/(12))

$\underset{r = 1, r \neq s}{\sum^{n}} \underset{s = 1}{\sum^{n}} \frac{rs}{n (n - 1)} \overset{?}{=} \frac{(n + 1) (3 n + 2)}{12}$

Question Number 151026 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (0^n /(n!))=?

$\underset{n = 0}{\sum^{\infty}} \frac{0^{n}}{n!} = ?$

Question Number 151053 Answers: 1 Comments: 0

Question Number 151017 Answers: 0 Comments: 0

∫_0 ^( ∞) (ln(x^(√x) −1))^(−x) dx = ?

$\int_{0}^{\infty} {(\ln (x^{\sqrt{x}} - 1))}^{- x} d x = ?$

Question Number 150994 Answers: 0 Comments: 0

show that the improper integral lim_(B→∞) ∫_0 ^( B) sin(x)sin(x^2 )dx converges

$show that the improper integral$ $\lim_{B \to \infty} \int_{0}^{B} \sin (x) \sin (x^{2}) d x$ $converges$

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