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IntegrationQuestion and Answers: Page 127

Question Number 118928 Answers: 1 Comments: 2

Question Number 118927 Answers: 0 Comments: 0

Question Number 118924 Answers: 1 Comments: 0

... advanced calculus... prove that :: Σ_(n=1) ^∞ (((−1)^n H_n )/n^2 ) =∫_0 ^( 1) ((ln(1−x)ln(1+x) )/x)dx note :: H_n =Σ_(k=1) ^n (1/k) therefore: Σ_(n=1 ) ^∞ (((−1)^n H_n )/n^2 )=((−5)/8) ζ (3 ) ✓ ..m.n.july.1970...

$. . . a d v a n c e d c a l c u l u s . . .$ $p r o v e t h a t ::$ $\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} H_{n}}{n^{2}} = \int_{0}^{1} \frac{l n (1 - x) l n (1 + x)}{x} d x$ $n o t e :: H_{n} = \underset{k = 1}{\sum^{n}} \frac{1}{k}$ $t h e r e f o r e : \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} H_{n}}{n^{2}} = \frac{- 5}{8} ζ (3) ✓$ $. . m . n . j u l y . 1970 . . .$

Question Number 118905 Answers: 4 Comments: 0

∫ (dλ/((λ^2 −9)^2 )) =?

$\int \frac{d λ}{{(λ^{2} - 9)}^{2}} = ?$

Question Number 118891 Answers: 2 Comments: 0

∫_0 ^π arctan(3^(cosx) )dx=??? please help

$\int_{0}^{π} a r c t a n (3^{c o s x}) d x = ? ? ?$ $p l e a s e h e l p$

Question Number 118886 Answers: 1 Comments: 0

... advanced calculus... evaluate :: {_(2. Ω_2 = ∫_0 ^( (1/2)) ((ln^2 (1+x))/x) dx=??) ^(1. Ω_1 =∫_0 ^( (1/2)) ((ln^2 (1−x))/x)dx=??) ... M.N.1970...

$. . . a d v a n c e d c a l c u l u s . . .$ $e v a l u a t e :: {_{2 . Ω_{2} = \int_{0}^{\frac{1}{2}} \frac{{l n}^{2} (1 + x)}{x} d x = ? ?}^{1 . Ω_{1} = \int_{0}^{\frac{1}{2}} \frac{{l n}^{2} (1 - x)}{x} d x = ? ?}$ $. . . M . N . 1970 . . .$

Question Number 118834 Answers: 2 Comments: 0

∫ ((x^4 +1)/(x^5 +4x^3 )) dx

$\int \frac{x^{4} + 1}{x^{5} + 4 x^{3}} d x$

Question Number 118819 Answers: 3 Comments: 0

∫((x^4 +x^2 +1)/((x^2 +4)^2 (x^2 +1))) dx

$\int \frac{x^{4} + x^{2} + 1}{{(x^{2} + 4)}^{2} (x^{2} + 1)} d x$

Question Number 118753 Answers: 3 Comments: 1

∫_2 ^4 x^3 e^x dx

$\int_{2}^{4} x^{3} e^{x} d x$

Question Number 118752 Answers: 0 Comments: 0

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

$\int \frac{2 d x}{x^{2} \sqrt[4]{{(3 + x^{4})}^{5}}} d x$

Question Number 118710 Answers: 2 Comments: 0

Question Number 118705 Answers: 2 Comments: 0

... ⧫Advanced Calculus⧫... Evaluate:: Ω = ∫_0 ^( 1 ) ((sin^(−1) (x))/(1+x^2 ))dx ...♠L𝛗rD ∅sE♠... ...♣GooD LucK♣

$. . . ⧫ Advanced Calculus ⧫ . . .$ $Evaluate ::$ $Ω = \int_{0}^{1} \frac{\sin^{- 1} (x)}{1 + x^{2}} dx$ $. . . ♠ L ϕ rD \emptyset sE ♠ . . .$ $. . . ♣ GooD LucK ♣$

Question Number 118668 Answers: 1 Comments: 0

∫ ((x∣sin x∣)/(1+cos^2 x)) dx ?

$\int \frac{x ∣ \sin x ∣}{1 + \cos^{2} x} d x ?$

Question Number 118679 Answers: 2 Comments: 0

Question Number 118663 Answers: 1 Comments: 2

Given f(x) = ∫ _0 ^x (dt/([f(t)]^2 )) and ∫ _0 ^2 (dt/([ f(t)]^2 )) = (6)^(1/(3 )) Then then the value of f(9) is __

$G i v e n f (x) = \int \underset{0}{\overset{x}{}} \frac{d t}{{[f (t)]}^{2}} a n d \int \underset{0}{\overset{2}{}} \frac{d t}{{[f (t)]}^{2}} = \sqrt[3]{6}$ $T h e n t h e n t h e v a l u e o f f (9) i s__$

Question Number 118575 Answers: 1 Comments: 0

∫ cos (2cot^(−1) (√((1−x)/(1+x))) ) dx ?

$\int \cos ({2 \cot}^{- 1} \sqrt{\frac{1 - x}{1 + x}}) d x ?$

Question Number 118566 Answers: 6 Comments: 2

solve ∫_0 ^1 (dx/( (√x) (√(1−x)) )) .

$s o l v e \underset{0}{\int^{1}} \frac{d x}{\sqrt{x} \sqrt{1 - x}} .$

Question Number 118541 Answers: 3 Comments: 1

∫ tan (x).tan (2x).tan (3x) dx = ?

$\int \tan (x) . \tan (2 x) . \tan (3 x) d x = ?$

Question Number 121164 Answers: 0 Comments: 0

...ADVANCED CALCULUS... If ∫_0 ^( ∞) ln(x)sin(x^2 )dx =λ∫_0 ^( ∞) sin(x^2 )dx then find the value of ′′λ′′ . ...m.n.july.1970...

$. . . ADVANCED CALCULUS . . .$ $If \int_{0}^{\infty} l n (x) s i n (x^{2}) d x = λ \int_{0}^{\infty} s i n (x^{2}) d x$ $t h e n f i n d t h e v a l u e o f^{″} λ^{″} .$ $. . . m . n . july . 1970 . . .$

Question Number 118491 Answers: 1 Comments: 0

... ⧫Advanced Calculus⧫... Evaluate:: Ω = ∫_0 ^( ∞) ((secθ)/( (√(4tan^2 θ+5))))dθ ...♠L𝛗rD ∅sE♠... ...♣GooD LucK♣

$. . . ⧫ Advanced Calculus ⧫ . . .$ $Evaluate ::$ $Ω = \int_{0}^{\infty} \frac{\sec θ}{\sqrt{{4 \tan}^{2} θ + 5}} d θ$ $. . . ♠ L ϕ rD \emptyset sE ♠ . . .$ $. . . ♣ GooD LucK ♣$

Question Number 118478 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−2x) ln(1+3x)dx

$find \int_{0}^{\infty} e^{- 2 x} \ln (1 + 3 x) dx$

Question Number 118476 Answers: 0 Comments: 0

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

$find \int_{- \infty}^{\infty} \frac{\arctan (1 + 2 x)}{x^{2} + 1} dx$

Question Number 118475 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(2+x^2 ))/(x^2 +9))dx

$calculate \int_{0}^{\infty} \frac{\arctan (2 + x^{2})}{x^{2} + 9} dx$

Question Number 118438 Answers: 0 Comments: 0

... nice calculus... evaluate :: lim_(s→0) ((ζ( 1+s )+ζ(1−s))/2) =^? γ γ: euler−mascheroni constant m.n.1970.

$. . . n i c e c a l c u l u s . . .$ $e v a l u a t e ::$ ${l i m}_{s \to 0} \frac{ζ (1 + s) + ζ (1 - s)}{2} \overset{?}{=} γ$ $γ : e u l e r - m a s c h e r o n i c o n s t a n t$ $m . n . 1970 .$

Question Number 118436 Answers: 4 Comments: 0

∫ cos^4 (x) cos^4 (2x) dx

$\int \cos^{4} (x) \cos^{4} (2 x) d x$

Question Number 118419 Answers: 4 Comments: 1

∫ ((2sin 2x)/(4cos x+sin 2x)) dx

$\int \frac{2 \sin 2 x}{4 \cos x + \sin 2 x} d x$

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