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IntegrationQuestion and Answers: Page 292

Question Number 34295 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−n[x]) cos(x)dx with n>0

$f i n d \int_{0}^{\infty} e^{- n [x]} c o s (x) d x w i t h n > 0$

Question Number 34294 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−nx) ∣sinx∣dx with n>0

$f i n d \int_{0}^{\infty} e^{- n x} ∣ s i n x ∣ d x w i t h n > 0$

Question Number 34293 Answers: 0 Comments: 0

calculate ∫∫_D x^2 y dxdy? with D = {(x,y)∈ R^2 / 0≤y≤1−x^2 ,∣x+y +3∣ ≤5}

$c a l c u l a t e \int \int_{D} x^{2} y d x d y ? w i t h$ $D = {(x, y) \in R^{2} / 0 ⩽ y ⩽ 1 - x^{2}, ∣ x + y + 3 ∣ ⩽ 5}$

Question Number 34292 Answers: 0 Comments: 1

calculate ∫∫_w (x+y)e^(x−y) dxdy with w={(x,y)∈R^2 / ∣x∣ ≤1 and ∣y+1∣≤3 }

$c a l c u l a t e \int \int_{w} (x + y) e^{x - y} d x d y w i t h$ $w = {(x, y) \in R^{2} / ∣ x ∣ ⩽ 1 a n d ∣ y + 1 ∣⩽ 3}$

Question Number 34291 Answers: 0 Comments: 0

let B(x,y) = ∫_0 ^1 u^(x−1) (1−u)^(y−1) du and Γ(x)= ∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x) = 2∫_0 ^∞ u^(2x−1) e^(−u^2 ) du 2)give Γ(x)Γ(y) at form of double integrale 3)prove that B(x,y) =((Γ(x)Γ(y))/(Γ(x+y))) 4) calculate B(m,n) for m and n integr naturals

$l e t B (x, y) = \int_{0}^{1} u^{x - 1} {(1 - u)}^{y - 1} d u a n d$ $Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t$ $1) p r o v e t h a t Γ (x) = 2 \int_{0}^{\infty} u^{2 x - 1} e^{- u^{2}} d u$ $2) g i v e Γ (x) Γ (y) a t f o r m o f d o u b l e i n t e g r a l e$ $3) p r o v e t h a t B (x, y) = \frac{Γ (x) Γ (y)}{Γ (x + y)}$ $4) c a l c u l a t e B (m, n) f o r m a n d n i n t e g r n a t u r a l s$

Question Number 34290 Answers: 0 Comments: 0

calculate ∫∫_D ((dxdy)/((1+x+y)^2 )) D ={(x,y)∈ R^2 / 1≤x+y≤ 2}

$c a l c u l a t e \int \int_{D} \frac{d x d y}{{(1 + x + y)}^{2}}$ $D = {(x, y) \in R^{2} / 1 ⩽ x + y ⩽ 2}$

Question Number 34289 Answers: 0 Comments: 1

calculate ∫∫_w (xy −2)dxdy with w = {(x,y)∈R^2 / x≥0 and 1≤y≤2−x }

$c a l c u l a t e \int \int_{w} (x y - 2) d x d y w i t h$ $w = {(x, y) \in R^{2} / x ⩾ 0 a n d 1 ⩽ y ⩽ 2 - x}$

Question Number 34288 Answers: 0 Comments: 0

calculate ∫∫_w e^(−yx^2 ) (x+y)dxdy with w =[0,1]^2

$c a l c u l a t e \int \int_{w} e^{- {y x}^{2}} (x + y) d x d y w i t h$ $w = {[0, 1]}^{2}$

Question Number 34287 Answers: 0 Comments: 0

calculate ∫∫_D xydxdy with D={(x,y)∈R^2 /x≥0 ,y≥0 , x+y ≤ (3/2)}

$c a l c u l a t e \int \int_{D} x y d x d y w i t h$ $D = {(x, y) \in R^{2} / x ⩾ 0, y ⩾ 0, x + y ⩽ \frac{3}{2}}$

Question Number 34286 Answers: 0 Comments: 2

find ∫ ((artanx)/((1+x)^2 ))dx

$f i n d \int \frac{a r t a n x}{{(1 + x)}^{2}} d x$

Question Number 34285 Answers: 0 Comments: 3

find ∫ (dx/((1+chx)^2 )) 2) calculate ∫_0 ^1 (dx/((1+chx)^2 ))

$f i n d \int \frac{d x}{{(1 + c h x)}^{2}}$ $2) c a l c u l a t e \int_{0}^{1} \frac{d x}{{(1 + c h x)}^{2}}$

Question Number 34284 Answers: 0 Comments: 1

find ∫ (dt/(sin(2t)))

$f i n d \int \frac{d t}{s i n (2 t)}$

Question Number 34283 Answers: 0 Comments: 2

calculate ∫_0 ^(π/2) (dx/(cos^4 x +sin^4 x))

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d x}{{c o s}^{4} x + {s i n}^{4} x}$

Question Number 34282 Answers: 0 Comments: 1

find ∫_(π/6) ^(π/3) (dx/(cos(x) sin(x)))

$f i n d \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{d x}{c o s (x) s i n (x)}$

Question Number 34281 Answers: 0 Comments: 0

calculate ∫_1 ^(√3) ((x−1)/(x^2 (x^2 +1)))dx

$c a l c u l a t e \int_{1}^{\sqrt{3}} \frac{x - 1}{x^{2} (x^{2} + 1)} d x$

Question Number 34280 Answers: 0 Comments: 0

find ∫ ((ln(x+x^2 ))/x^2 )dx

$f i n d \int \frac{l n (x + x^{2})}{x^{2}} d x$

Question Number 34279 Answers: 0 Comments: 0

find ∫_1 ^(+∞) (((−1)^([x]) )/x) dx .

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

Question Number 34278 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ (2 +(t+3)ln(((t+2)/(t+4))))dt .

$f i n d t h e v a l u e o f \int_{0}^{\infty} (2 + (t + 3) l n (\frac{t + 2}{t + 4})) d t .$

Question Number 34277 Answers: 0 Comments: 1

calculate A_n =∫_0 ^∞ (dx/((x+1)(x+2)....(x+n))) n integr≥2 .

$c a l c u l a t e A_{n} = \int_{0}^{\infty} \frac{d x}{(x + 1) (x + 2) . . . . (x + n)}$ $n i n t e g r ⩾ 2 .$

Question Number 34276 Answers: 0 Comments: 0

nature of ∫_0 ^∞ (dx/(1+x^3 sin^2 x)) ?

$n a t u r e o f \int_{0}^{\infty} \frac{d x}{1 + x^{3} {s i n}^{2} x} ?$

Question Number 34274 Answers: 0 Comments: 0

calculate ∫_2 ^(+∞) ((4x)/(x^4 −1))dx .

$c a l c u l a t e \int_{2}^{+ \infty} \frac{4 x}{x^{4} - 1} d x .$

Question Number 34273 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (e^(arctanx) /(1+x^2 ))dx .

$c a l c u l a t e \int_{0}^{\infty} \frac{e^{a r c t a n x}}{1 + x^{2}} d x .$

Question Number 34271 Answers: 0 Comments: 0

find lim_(n→+∞) ∫_0 ^∞ ((arctan(nx))/(n(1+x^2 )))dx

$f i n d {l i m}_{n \to + \infty} \int_{0}^{\infty} \frac{a r c t a n (n x)}{n (1 + x^{2})} d x$

Question Number 34270 Answers: 0 Comments: 0

let give A_n = ∫_0 ^∞ (dx/((1+x^3 )^n )) 1) calculate A_1 2) for n≥2 find a relation between A_(n+1) and A_n 3) find the value of A_n .

$l e t g i v e A_{n} = \int_{0}^{\infty} \frac{d x}{{(1 + x^{3})}^{n}}$ $1) c a l c u l a t e A_{1}$ $2) f o r n ⩾ 2 f i n d a r e l a t i o n b e t w e e n A_{n + 1} a n d A_{n}$ $3) f i n d t h e v a l u e o f A_{n} .$

Question Number 34269 Answers: 0 Comments: 0

calculate I(λ) =∫_0 ^∞ (dx/((1+x^2 )(1+x^λ )))

$c a l c u l a t e I (λ) = \int_{0}^{\infty} \frac{d x}{(1 + x^{2}) (1 + x^{λ})}$

Question Number 34268 Answers: 0 Comments: 0

calculate I = ∫_(−(π/2)) ^(π/2) ln(1+sinx)dx

$c a l c u l a t e I = \int_{- \frac{π}{2}}^{\frac{π}{2}} l n (1 + s i n x) d x$

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