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IntegrationQuestion and Answers: Page 296

Question Number 33895 Answers: 3 Comments: 0

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) prove that Γ(x)Γ(1−x)= (π/(sin(πx))) 2) find the value of ∫_0 ^∞ e^(−x^2 ) dx .

$l e t Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 0$ $1) p r o v e t h a t Γ (x) Γ (1 - x) = \frac{π}{s i n (π x)}$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- x^{2}} d x .$

Question Number 33894 Answers: 0 Comments: 1

1)let f R→C 2π periodic even /f(x)=x ∀ x∈[0,π[ developp f at fourier serie 2) calculate Σ_(p=0) ^∞ (1/((2p+1)^2 )) .

$1) l e t f R \to C 2 π p e r i o d i c e v e n / f (x) = x$ $\forall x \in [0, π [d e v e l o p p f a t f o u r i e r s e r i e$ $2) c a l c u l a t e \sum_{p = 0}^{\infty} \frac{1}{{(2 p + 1)}^{2}} .$

Question Number 33888 Answers: 0 Comments: 0

developp at integr serie f(x)= ∫_0 ^(π/2) (dt/(√(1−x^2 sin^2 t))) . with ∣x∣<1 .

$d e v e l o p p a t i n t e g r s e r i e f (x) = \int_{0}^{\frac{π}{2}} \frac{d t}{\sqrt{1 - x^{2} {s i n}^{2} t}} .$ $w i t h ∣ x ∣< 1 .$

Question Number 33885 Answers: 0 Comments: 1

developp at integr serie f(x)= ∫_0 ^x sin(t^2 )dt .

$d e v e l o p p a t i n t e g r s e r i e f (x) = \int_{0}^{x} s i n (t^{2}) d t .$

Question Number 33884 Answers: 0 Comments: 1

let F(x)= ∫_0 ^(π/2) ((arctan(xtant))/(tant)) dt find a simple form of f(x) . 2) find the value of ∫_0 ^(π/2) ((arctan(2tant))/(tant))dt .

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

Question Number 33883 Answers: 0 Comments: 1

find a simple form of f(x)=∫_0 ^(π/2) ln(1+xsin^2 t)dt with ∣x∣<1.

$f i n d a s i m p l e f o r m o f f (x) = \int_{0}^{\frac{π}{2}} l n (1 + {x s i n}^{2} t) d t$ $w i t h ∣ x ∣< 1 .$

Question Number 33845 Answers: 0 Comments: 1

let I_n = ∫_0 ^1 ((arctan(1 +n))/(√(1+x^n ))) find lim_(n→+∞) I_n .

$l e t I_{n} = \int_{0}^{1} \frac{a r c t a n (1 + n)}{\sqrt{1 + x^{n}}} f i n d {l i m}_{n \to + \infty} I_{n} .$

Question Number 33835 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cos(πx))/((x^2 +1+i)^2 )) dx

$f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{c o s (π x)}{{(x^{2} + 1 + i)}^{2}} d x$

Question Number 33787 Answers: 0 Comments: 0

lim_(n→∞) ((1/n) ∫_1 ^n n^(1/x) dx)

$\lim_{n \to \infty} (\frac{1}{n} \underset{1}{\int^{n}} n^{\frac{1}{x}} d x)$

Question Number 33823 Answers: 1 Comments: 0

solve : I = ∫_0 ^π (((r−R cosθ) sin θ )/((R^(2 ) + r^2 − 2Rr cos θ)^(3/2) )) dθ for r < R and r > R respectively.

$s o l v e :$ $I = \underset{0}{\int^{π}} \frac{(r - R c o s θ) s i n θ}{{(R^{2} + r^{2} - 2 R r c o s θ)}^{3 / 2}} d θ$ $f o r r < R$ $a n d r > R r e s p e c t i v e l y .$

Question Number 33759 Answers: 0 Comments: 9

solve : ∫_(−π/2) ^(π/2) ((sin θ )/(√( R^2 + r^2 − 2rR cos θ))) dθ

$s o l v e :$ $\underset{- π / 2}{\int^{π / 2}} \frac{s i n θ}{\sqrt{R^{2} + r^{2} - 2 r R c o s θ}} d θ$

Question Number 33747 Answers: 0 Comments: 0

Calculate ∫_(−∞) ^(+∞) e^(−x^2 ) dx using Residue theorem

$C a l c u l a t e \int_{- \infty}^{+ \infty} e^{- x^{2}} d x u s i n g R e s i d u e t h e o r e m$

Question Number 33744 Answers: 0 Comments: 1

let P_n (x)=(1+x^2 )(1+x^4 )....(1+x^2^n ) calculate lim_(n→+∞) ∫_0 ^x P_n (t)dt with 0<x<1 .

$l e t P_{n} (x) = (1 + x^{2}) (1 + x^{4}) . . . . (1 + x^{2^{n}})$ $c a l c u l a t e {l i m}_{n \to + \infty} \int_{0}^{x} P_{n} (t) d t w i t h 0 < x < 1 .$

Question Number 33737 Answers: 1 Comments: 3

find the value of ∫_0 ^∞ ((cos(xt))/((t^2 + x^2 )^2 )) dt .

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{c o s (x t)}{{(t^{2} + x^{2})}^{2}} d t .$

Question Number 33736 Answers: 2 Comments: 1

find the value of ∫_(−∞) ^(+∞) (x^2 /((1+x +x^2 )^2 ))dx

$f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{x^{2}}{{(1 + x + x^{2})}^{2}} d x$

Question Number 33735 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(2x)dx)/((x^2 +1)( 2x^2 +3))) .

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

Question Number 33705 Answers: 1 Comments: 1

let α>0 find the fourier transform of f(t) = e^(−a^2 t^2 )

$l e t α > 0 f i n d t h e f o u r i e r t r a n s f o r m o f$ $f (t) = e^{- a^{2} t^{2}}$

Question Number 33704 Answers: 0 Comments: 1

let f(t) = (1/(a^2 +t^2 )) witha>0 give the fourier transformfor f .

$l e t f (t) = \frac{1}{a^{2} + t^{2}} w i t h a > 0 g i v e t h e f o u r i e r$ $t r a n s f o r m f o r f .$

Question Number 33703 Answers: 0 Comments: 0

give ∫_0 ^∞ ((x e^(−x) )/(1 −e^(−2x) )) sin(πx)dx at form of serie.

$g i v e \int_{0}^{\infty} \frac{x e^{- x}}{1 - e^{- 2 x}} s i n (π x) d x a t f o r m o f s e r i e .$

Question Number 33695 Answers: 0 Comments: 1

find lim_(n→+∞) ∫_0 ^∞ (e^(−(x/n)) /(1+x^2 ))dx.

$f i n d {l i m}_{n \to + \infty} \int_{0}^{\infty} \frac{e^{- \frac{x}{n}}}{1 + x^{2}} d x .$

Question Number 33694 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^∞ (dx/(x^n +e^x )) .

$c a l c u l a t e {l i m}_{n \to + \infty} \int_{0}^{\infty} \frac{d x}{x^{n} + e^{x}} .$

Question Number 33689 Answers: 2 Comments: 1

∫(x/(x^3 +1))dx

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

Question Number 33677 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((xlnx)/(x−1))dx .

$c a l c u l a t e \int_{0}^{1} \frac{x l n x}{x - 1} d x .$

Question Number 33619 Answers: 1 Comments: 3

∫x^(5/2) (1−x)^(3/2) dx

$\int x^{5 / 2} {(1 - x)}^{3 / 2} d x$

Question Number 33599 Answers: 1 Comments: 2

calculatef(a)= ∫_(−a) ^a (dx/((t^2 +x^2 )^(3/2) )) with a>0 .

$c a l c u l a t e f (a) = \int_{- a}^{a} \frac{d x}{{(t^{2} + x^{2})}^{\frac{3}{2}}} w i t h a > 0 .$

Question Number 33590 Answers: 0 Comments: 1

let α >1 calculate f(α) = ∫_α ^(+∞) ((x^2 −x+1)/((x−1)^2 (x+1)^2 )) dx .

$l e t α > 1 c a l c u l a t e f (α) = \int_{α}^{+ \infty} \frac{x^{2} - x + 1}{{(x - 1)}^{2} {(x + 1)}^{2}} d x .$

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