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IntegrationQuestion and Answers: Page 306

Question Number 30777 Answers: 0 Comments: 0

find interms of n A_n = ∫_0 ^∞ ((ln(x))/((1+x^ )^n )) dx with n from N and n≥3 .

$f i n d i n t e r m s o f n A_{n} = \int_{0}^{\infty} \frac{l n (x)}{{(1 + x^{})}^{n}} d x w i t h n f r o m$ $N a n d n ⩾ 3 .$

Question Number 30776 Answers: 1 Comments: 1

find ∫_0 ^1 ((xdx)/((1+x^2 )(√(1−x^4 )))) .

$f i n d \int_{0}^{1} \frac{x d x}{(1 + x^{2}) \sqrt{1 - x^{4}}} .$

Question Number 30775 Answers: 1 Comments: 1

letα ∈]0,π[ calculate ∫_0 ^(π/2) (dx/(2(cosα +chx))) .

$l e t α \in] 0, π [c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d x}{2 (c o s α + c h x)} .$

Question Number 30774 Answers: 1 Comments: 1

find f(t)=∫_0 ^1 ln(1+tx^2 )dx with t>0

$f i n d f (t) = \int_{0}^{1} l n (1 + {t x}^{2}) d x w i t h t > 0$

Question Number 30773 Answers: 0 Comments: 1

let a>0 calculate ∫_0 ^∞ (dx/((x^2 +a^2 )^2 )) 2) calculate ∫_(−∞) ^(+∞) (x^2 /((x^2 +a^2 )^2 ))dx.

$l e t a > 0 c a l c u l a t e \int_{0}^{\infty} \frac{d x}{{(x^{2} + a^{2})}^{2}}$ $2) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{x^{2}}{{(x^{2} + a^{2})}^{2}} d x .$

Question Number 30769 Answers: 0 Comments: 1

find the value of I= ∫_0 ^1 (dx/((x+1)^2 (√(x^2 +2x +2)))) .

$f i n d t h e v a l u e o f I = \int_{0}^{1} \frac{d x}{{(x + 1)}^{2} \sqrt{x^{2} + 2 x + 2}} .$

Question Number 30767 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ((1−(√x))/(1−^n (√x)))dx.

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

Question Number 30766 Answers: 0 Comments: 0

find ∫_0 ^1 (x/(√(x^4 +x^2 +1)))dx

$f i n d \int_{0}^{1} \frac{x}{\sqrt{x^{4} + x^{2} + 1}} d x$

Question Number 30765 Answers: 0 Comments: 1

find ∫_0 ^1 (dx/(√(x^2 +x+1))) .

$f i n d \int_{0}^{1} \frac{d x}{\sqrt{x^{2} + x + 1}} .$

Question Number 30764 Answers: 0 Comments: 1

let I_n = ∫_0 ^(1/2) (1−2t)^n e^(−t) dt with n integr not 0 1) prove that ∀t∈[0,(1/2)] (1/(√e))(1−2t)^n ≤ (1−2t)^n e^(−t) ≤(1−2t)^n then find lim_(n→∞ ) I_n 2) prove that I_(n+1 ) =1−2(n+1)I_n 3) calculate I_1 ,I_2 , and I_3 .

$l e t I_{n} = \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} e^{- t} d t w i t h n i n t e g r n o t 0$ $1) p r o v e t h a t \forall t \in [0, \frac{1}{2}]$ $\frac{1}{\sqrt{e}} {(1 - 2 t)}^{n} ⩽ {(1 - 2 t)}^{n} e^{- t} ⩽ {(1 - 2 t)}^{n} t h e n f i n d {l i m}_{n \to \infty} I_{n}$ $2) p r o v e t h a t I_{n + 1} = 1 - 2 (n + 1) I_{n}$ $3) c a l c u l a t e I_{1}, I_{2}, a n d I_{3} .$

Question Number 30761 Answers: 0 Comments: 1

find ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx with n from N.

$f i n d \int_{0}^{\infty} x^{2 n + 1} e^{- x^{2}} d x w i t h n f r o m N .$

Question Number 30760 Answers: 0 Comments: 1

find I_n = ∫_0 ^1 (lnx)^n dx with n fromN

$f i n d I_{n} = \int_{0}^{1} {(l n x)}^{n} d x w i t h n f r o m N$

Question Number 30741 Answers: 0 Comments: 0

let give D= R_+ ^2 −{(0,0)} and α from R let C_1 ={(x,y)∈ D/0<x^2 +y^2 ≤1 } C_2 ={(x,y) ∈D / x^2 +y^2 ≥1} study the convergence of I= ∫∫_C_1 ((dxdy)/(((√(x^2 +y^2 )) )^α )) and J=∫∫_C_2 ((dxdy)/(((√(x^2 +y^2 )) )^α )) .

$l e t g i v e D = R_{+}^{2} - {(0, 0)} a n d α f r o m R l e t$ $C_{1} = {(x, y) \in D / 0 < x^{2} + y^{2} ⩽ 1}$ $C_{2} = {(x, y) \in D / x^{2} + y^{2} ⩾ 1} s t u d y t h e c o n v e r g e n c e o f$ $I = \int \int_{C_{1}} \frac{d x d y}{{(\sqrt{x^{2} + y^{2}})}^{α}} a n d J = \int \int_{C_{2}} \frac{d x d y}{{(\sqrt{x^{2} + y^{2}})}^{α}} .$

Question Number 30737 Answers: 0 Comments: 1

∫(1/(x^2 +ln x))dx

$\int \frac{1}{x^{2} + \ln x} d x$

Question Number 30665 Answers: 0 Comments: 0

find ∫_0 ^π (dx/(1+cos(2x) +sin(2x))) .

$f i n d \int_{0}^{π} \frac{d x}{1 + c o s (2 x) + s i n (2 x)} .$

Question Number 30585 Answers: 0 Comments: 0

find F_n (x)= ∫_0 ^∞ (x^n /(e^(x+n) +1))dx .

$f i n d F_{n} (x) = \int_{0}^{\infty} \frac{x^{n}}{e^{x + n} + 1} d x .$

Question Number 30584 Answers: 0 Comments: 0

find I= ∫_(−∞) ^(+∞) (e^(−x^2 ) /(a^2 +(v−x)^2 ))dx.

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

Question Number 30580 Answers: 0 Comments: 1

decompose inside C[x] F= (x^n /(x^m +1)) with m≥n+2 then find ∫_0 ^∞ (x^n /(x^m +1))dx.

$d e c o m p o s e i n s i d e C [x] F = \frac{x^{n}}{x^{m} + 1} w i t h m ⩾ n + 2$ $t h e n f i n d \int_{0}^{\infty} \frac{x^{n}}{x^{m} + 1} d x .$

Question Number 30575 Answers: 0 Comments: 0

find ∫∫_D (x^2 +y^2 )dxdy with D={(x,y)/ x≤1 and x^2 ≤y≤2 }.

$f i n d \int \int_{D} (x^{2} + y^{2}) d x d y w i t h$ $D = {(x, y) / x ⩽ 1 a n d x^{2} ⩽ y ⩽ 2} .$

Question Number 30574 Answers: 0 Comments: 0

find ∫∫_([1,e]^2 ) ln(xy)dxdy.

$f i n d \int \int_{{[1, e]}^{2}} l n (x y) d x d y .$

Question Number 30573 Answers: 0 Comments: 0

find ∫∫_([0,1]×[0,1]) (x^2 /(1+y^2 ))dxdy.

$f i n d \int \int_{[0, 1] \times [0, 1]} \frac{x^{2}}{1 + y^{2}} d x d y .$

Question Number 30572 Answers: 0 Comments: 1

find I=∫∫_([3,4]×[1,2]) ((dxdy)/((x+y)^2 )) .

$f i n d I = \int \int_{[3, 4] \times [1, 2]} \frac{d x d y}{{(x + y)}^{2}} .$

Question Number 30570 Answers: 0 Comments: 0

find ∫∫_U ((dxdy)/(x^2 +y^2 )) with U= {(x,y)∈R^2 /1≤x^2 +2y^2 ≤4}

$f i n d \int \int_{U} \frac{d x d y}{x^{2} + y^{2}} w i t h U = {(x, y) \in R^{2} / 1 ⩽ x^{2} + 2 y^{2} ⩽ 4}$

Question Number 30569 Answers: 0 Comments: 0

find I= ∫∫_D (√(1−(x^2 /a^2 )−(y^2 /b^2 ))) dxdy with D is the interior of ellipce (x^2 /a^2 ) +(y^2 /b^2 ) =1.

$f i n d I = \int \int_{D} \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}} d x d y w i t h D i s t h e i n t e r i o r$ $o f e l l i p c e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 .$

Question Number 30568 Answers: 0 Comments: 0

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

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

Question Number 30566 Answers: 0 Comments: 0

study the convergence of A(α) =∫_0 ^∞ (t^(α−1) /(1+t^2 ))dt and find its value.

$s t u d y t h e c o n v e r g e n c e o f A (α) = \int_{0}^{\infty} \frac{t^{α - 1}}{1 + t^{2}} d t a n d$ $f i n d i t s v a l u e .$

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