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IntegrationQuestion and Answers: Page 125

Question Number 122290 Answers: 1 Comments: 1

... nice calculus... calculate :: Ω=∫_0 ^( 1) x^2 (ψ(1+x)−ψ(2−x))dx=??? .m.n.1970.

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

Question Number 122274 Answers: 2 Comments: 0

∫ ((√(1−(√x)))/( (√(1+(√x))))) dx ?

$\int \frac{\sqrt{1 - \sqrt{x}}}{\sqrt{1 + \sqrt{x}}} d x ?$

Question Number 122205 Answers: 1 Comments: 2

find ∫ (dx/(x(x+1)(√(x^2 +x))))

$find \int \frac{dx}{x (x + 1) \sqrt{x^{2} + x}}$

Question Number 122186 Answers: 2 Comments: 0

Obtain a reduction formulae for I_n = ∫_0 ^1 (ln x)^n dx find I_2 = ∫_0 ^1 (ln x)^2 dx

$Obtain a reduction formulae for$ $I_{n} = \underset{0}{\int^{1}} {(\ln x)}^{n} d x$ $find I_{2} = \underset{0}{\int^{1}} {(\ln x)}^{2} d x$

Question Number 122160 Answers: 1 Comments: 0

Question Number 122159 Answers: 3 Comments: 0

... nice calculus... prove that:: Ω=∫_0 ^( (π/2)) {tan^(−1) (ptan(x))−tan^(−1) (qtan(x))}(tan(x)+cot(x))dx =(π/2) log((p/q)) ( p , q >0 ) m.n.

$. . . n i c e c a l c u l u s . . .$ $p r o v e t h a t ::$ $Ω = \int_{0}^{\frac{π}{2}} {{t a n}^{- 1} (p t a n (x)) - {t a n}^{- 1} (q t a n (x))} (t a n (x) + c o t (x)) d x$ $= \frac{π}{2} l o g (\frac{p}{q}) (p, q > 0)$ $m . n .$

Question Number 122157 Answers: 2 Comments: 1

find ∫_(−1) ^1 (√(1+x^4 ))dx

$find \int_{- 1}^{1} \sqrt{1 + x^{4}} dx$

Question Number 122137 Answers: 1 Comments: 0

Question Number 122120 Answers: 2 Comments: 0

∫_0 ^3 (1/( (√y))).e^y dy

$\underset{0}{\int^{3}} \frac{1}{\sqrt{y}} . e^{y} d y$

Question Number 122109 Answers: 2 Comments: 0

Question Number 122108 Answers: 1 Comments: 0

∫_1 ^2 ((ln (x))/x^2 ) dx ?

$\underset{1}{\int^{2}} \frac{\ln (x)}{x^{2}} d x ?$

Question Number 122098 Answers: 1 Comments: 1

Question Number 122044 Answers: 0 Comments: 3

Question Number 122035 Answers: 1 Comments: 0

...nice calculus... prove that : ∫_0 ^( 1) ((ln^2 (1+x))/x)dx=^(??) ((ζ(3))/4) .m.n.

$. . . n i c e c a l c u l u s . . .$ $p r o v e t h a t :$ $\int_{0}^{1} \frac{{l n}^{2} (1 + x)}{x} d x \overset{? ?}{=} \frac{ζ (3)}{4}$ $. m . n .$

Question Number 121972 Answers: 1 Comments: 0

∫_0 ^( ∞) e^(−x^2 ) cos(5x)dx

$\int_{0}^{\infty} e^{- x^{2}} \cos (5 x) dx$

Question Number 121965 Answers: 2 Comments: 0

∫_4 ^9 ((ℓn (x))/( (√x))) dx ?

$\underset{4}{\int^{9}} \frac{ℓ n (x)}{\sqrt{x}} d x ?$

Question Number 121962 Answers: 2 Comments: 0

∫_0 ^1 (dx/( ((x^2 −x^3 ))^(1/(3 )) )) ?

$\underset{0}{\int^{1}} \frac{d x}{\sqrt[3]{x^{2} - x^{3}}} ?$

Question Number 121928 Answers: 1 Comments: 2

Σ_(n=0) ^∞ ((1/(12n+1))+(1/(12n+5))−(1/(12n+7))−(1/(12n+11))) Problem source : Brilliant.Org

$\underset{n = 0}{\sum^{\infty}} (\frac{1}{12 n + 1} + \frac{1}{12 n + 5} - \frac{1}{12 n + 7} - \frac{1}{12 n + 11})$ $P r o b l e m s o u r c e : B r i l l i a n t . O r g$

Question Number 121887 Answers: 2 Comments: 2

Question Number 121886 Answers: 2 Comments: 1

Question Number 121875 Answers: 2 Comments: 0

Question Number 121862 Answers: 1 Comments: 0

1)explicite f(a)=∫_0 ^∞ ((t^(a−1) lnt)/(1+t))dt with 0<a<1 2)calculate ∫_0 ^∞ ((lnt)/((1+t)(√t)))dt

$1) e x p l i c i t e f (a) = \int_{0}^{\infty} \frac{t^{a - 1} l n t}{1 + t} d t$ $w i t h 0 < a < 1$ $2) c a l c u l a t e \int_{0}^{\infty} \frac{l n t}{(1 + t) \sqrt{t}} d t$

Question Number 121860 Answers: 0 Comments: 0

calculate ∫∫_([0,1]^2 ) ((arctan((√(x^2 +y^2 ))))/(x^2 +y^2 ))dxdy

$c a l c u l a t e \int \int_{{[0, 1]}^{2}} \frac{a r c t a n (\sqrt{x^{2} + y^{2}})}{x^{2} + y^{2}} d x d y$

Question Number 121859 Answers: 2 Comments: 0

find ∫ (dx/((x−1)(√(x+1))−(x+1)(√(x−1))))

$f i n d \int \frac{d x}{(x - 1) \sqrt{x + 1} - (x + 1) \sqrt{x - 1}}$

Question Number 121858 Answers: 0 Comments: 0

decompose F(x) =(x^n /(x^(2n+1) +1)) 2)find the value of ∫_0 ^∞ (x^n /(x^(2n+1) +1))dx n integr natural

$d e c o m p o s e F (x) = \frac{x^{n}}{x^{2 n + 1} + 1}$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{x^{n}}{x^{2 n + 1} + 1} d x$ $n i n t e g r n a t u r a l$

Question Number 121857 Answers: 0 Comments: 0

find ∫_0 ^∞ xe^(−x^2 ) arctan(2x)dx

$f i n d \int_{0}^{\infty} {x e}^{- x^{2}} a r c t a n (2 x) d x$

Pg 120 Pg 121 Pg 122 Pg 123 Pg 124 Pg 125 Pg 126 Pg 127 Pg 128 Pg 129

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