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IntegrationQuestion and Answers: Page 310

Question Number 29078 Answers: 0 Comments: 2

let give h(x)= ∫_0 ^1 ((arctan(xt))/(1+t^2 )) find h(x) .

$l e t g i v e h (x) = \int_{0}^{1} \frac{a r c t a n (x t)}{1 + t^{2}} f i n d h (x) .$

Question Number 29077 Answers: 0 Comments: 1

let give g(x)=∫_0 ^∞ ((arctan(x(1+t^2 )))/(1+t^2 ))dt find a simple form of g^′ (x) without integral.

$l e t g i v e g (x) = \int_{0}^{\infty} \frac{a r c t a n (x (1 + t^{2}))}{1 + t^{2}} d t f i n d a s i m p l e$ $f o r m o f g^{^{'}} (x) w i t h o u t i n t e g r a l .$

Question Number 29076 Answers: 0 Comments: 1

let give f(x)= ∫_0 ^1 ((arctan(x(1+t^2 )))/(1+t^2 ))dt find asimple form of f(x) without integral.

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

Question Number 29043 Answers: 0 Comments: 1

∫tan^− (1−sinx/1+sinx) dx

$\int \tan^{-} (1 - sinx / 1 + sinx) dx$

Question Number 29038 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((cos(at))/(1+t^4 ))dt.

$f i n d \int_{- \infty}^{+ \infty} \frac{c o s (a t)}{1 + t^{4}} d t .$

Question Number 29028 Answers: 0 Comments: 0

for t>0 and f(t)= (4πt)^(−(n/2)) e^(−(x^2 /(4t))) prove that ∫_R f_t (x)dx=1 ∀t>0.

$f o r t > 0 a n d f (t) = {(4 π t)}^{- \frac{n}{2}} e^{- \frac{x^{2}}{4 t}} p r o v e t h a t$ $\int_{R} f_{t} (x) d x = 1 \forall t > 0 .$

Question Number 29027 Answers: 0 Comments: 0

find ∫∫_D e^(−y) sin(2xy)dxdy with D=[0,1]×[0,+∞[ then find the value of ∫_0 ^∞ ((sin^2 t)/t) e^(−t) dt .

$f i n d \int \int_{D} e^{- y} s i n (2 x y) d x d y w i t h D = [0, 1] \times [0, + \infty [$ $t h e n f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{{s i n}^{2} t}{t} e^{- t} d t .$

Question Number 29018 Answers: 0 Comments: 0

∫ (√(Σ_(n = 0) ^∞ [(−1)^n tan^(2n) (2x)])) dx

$\int \sqrt{\underset{n = 0}{\sum^{\infty}} [{(- 1)}^{n} \tan^{2 n} (2 x)]} d x$

Question Number 29003 Answers: 1 Comments: 1

find ∫_0 ^∞ (dx/(1+x^3 )) .

$f i n d \int_{0}^{\infty} \frac{d x}{1 + x^{3}} .$

Question Number 29002 Answers: 0 Comments: 0

let give 0<p<1 calculate K(p)= ∫_(−∞) ^(+∞) (e^(pt) /(1+e^t ))dt.

$l e t g i v e 0 < p < 1 c a l c u l a t e K (p) = \int_{- \infty}^{+ \infty} \frac{e^{p t}}{1 + e^{t}} d t .$

Question Number 29001 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((cos(ξt))/(1+t^4 ))dt.

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{c o s (ξ t)}{1 + t^{4}} d t .$

Question Number 29000 Answers: 0 Comments: 1

prove thst ∫_R (e^(iξx) /(1+x^2 ))dx= π e^(−∣ξ∣) .

$p r o v e t h s t \int_{R} \frac{e^{i ξ x}}{1 + x^{2}} d x = π e^{- ∣ ξ ∣} .$

Question Number 28999 Answers: 0 Comments: 1

prove that ∫_0 ^∞ (e^(−t) /(√t))dt= e^(i(π/4)) ∫_0 ^∞ (e^(−ix) /(√x))dx.

$p r o v e t h a t \int_{0}^{\infty} \frac{e^{- t}}{\sqrt{t}} d t = e^{i \frac{π}{4}} \int_{0}^{\infty} \frac{e^{- i x}}{\sqrt{x}} d x .$

Question Number 28998 Answers: 0 Comments: 0

find ∫_γ (e^z /(z(z+1)))dz with γ={z∈C/ ∣z−1∣=2}

$f i n d \int_{γ} \frac{e^{z}}{z (z + 1)} d z w i t h γ = {z \in C / ∣ z - 1 ∣= 2}$

Question Number 28997 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) (dx/((1+x^2 )( 2+e^(ix) ))) .

$f i n d \int_{- \infty}^{+ \infty} \frac{d x}{(1 + x^{2}) (2 + e^{i x})} .$

Question Number 28996 Answers: 0 Comments: 0

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

$f i n d \int_{- \infty}^{+ \infty} \frac{x^{2}}{{(x^{2} + 1)}^{2} (x^{2} + 2 x + 2)} d x .$

Question Number 28995 Answers: 0 Comments: 0

find ∫_0 ^(2π) ((cos(2t))/(3−cost)) dt.

$f i n d \int_{0}^{2 π} \frac{c o s (2 t)}{3 - c o s t} d t .$

Question Number 28994 Answers: 0 Comments: 0

find A_n = ∫_(−∞) ^(+∞) (dx/((1+x^2 )^n )) with n from N and n≥1.

$f i n d A_{n} = \int_{- \infty}^{+ \infty} \frac{d x}{{(1 + x^{2})}^{n}} w i t h n f r o m N a n d n ⩾ 1 .$

Question Number 28993 Answers: 1 Comments: 0

L means laplacr trsnsform find L (sin(at)) and L(cos(at)).

$L m e a n s l a p l a c r t r s n s f o r m f i n d L (s i n (a t))$ $a n d L (c o s (a t)) .$

Question Number 28992 Answers: 0 Comments: 0

L means laplace transform find L(e^(at) )(s).

$L m e a n s l a p l a c e t r a n s f o r m f i n d L (e^{a t}) (s) .$

Question Number 28990 Answers: 0 Comments: 0

calculate ∫_γ (e^z /((z−1)(z+3)^2 ))dz with γ id the positif circle γ={z∈C/ ∣z∣=(3/2)}.

$c a l c u l a t e \int_{γ} \frac{e^{z}}{(z - 1) {(z + 3)}^{2}} d z w i t h γ i d t h e p o s i t i f$ $c i r c l e γ = {z \in C / ∣ z ∣= \frac{3}{2}} .$

Question Number 28989 Answers: 0 Comments: 1

find ∫_0 ^∞ ((sin^2 (3x))/x^2 )dx.

$f i n d \int_{0}^{\infty} \frac{{s i n}^{2} (3 x)}{x^{2}} d x .$

Question Number 28988 Answers: 0 Comments: 0

let give 0<α<1 find in terms of α the value of integral ∫_0 ^∞ (dx/(x^α (1+x))) .

$l e t g i v e 0 < α < 1 f i n d i n t e r m s o f α t h e v a l u e o f i n t e g r a l$ $\int_{0}^{\infty} \frac{d x}{x^{α} (1 + x)} .$

Question Number 28986 Answers: 0 Comments: 0

let give a>1 find ∫_0 ^(2π) (dt/(a+cost)) .

$l e t g i v e a > 1 f i n d \int_{0}^{2 π} \frac{d t}{a + c o s t} .$

Question Number 28985 Answers: 0 Comments: 0

let give I_(m,a) =∫_0 ^∞ ((cos(mx))/((1+x^2 )(x^2 +a^2 )))dx 1)verify that I_(m,1) =lim_(a→1) I_(m,a) 2) find the value of ∫_0 ^∞ ((x sin(mx))/((1+x^2 )^2 ))dx

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

Question Number 28984 Answers: 0 Comments: 0

find F( (1/(1+x^4 ))) F means fourier transform.

$f i n d F (\frac{1}{1 + x^{4}}) F m e a n s f o u r i e r t r a n s f o r m .$

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