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IntegrationQuestion and Answers: Page 317

Question Number 27757 Answers: 1 Comments: 0

calculate I= ∫_0 ^(π/2) (dx/(1+cosx)) and J= ∫_0^ ^(π/2) ((cosx)/(1+cosx))dx .

$c a l c u l a t e I = \int_{0}^{\frac{π}{2}} \frac{d x}{1 + c o s x} a n d J = \int_{0^{}}^{\frac{π}{2}} \frac{c o s x}{1 + c o s x} d x .$

Question Number 28038 Answers: 0 Comments: 0

let give f(x)=(√(x+y)) +1 and D={(x,y)∈R^2 / 0≤x≤1 and −1≤y≤1} find the value of ∫∫ f(x,y)dxdy .

$l e t g i v e f (x) = \sqrt{x + y} + 1 a n d D = {(x, y) \in R^{2} / 0 ⩽ x ⩽ 1$ $a n d - 1 ⩽ y ⩽ 1} f i n d t h e v a l u e o f \int \int f (x, y) d x d y .$

Question Number 27684 Answers: 0 Comments: 1

1) prove the existence of the integral I=∫_0 ^(π/2) ((ln(1+cosx))/(cosx))dx 2)prove that I= ∫∫_D ((siny)/(1+cosx cosy))dxdy with D=[0,(π/2)]^2 3)find the value of I.

$1) p r o v e t h e e x i s t e n c e o f t h e i n t e g r a l$ $I = \int_{0}^{\frac{π}{2}} \frac{l n (1 + c o s x)}{c o s x} d x$ $2) p r o v e t h a t I = \int \int_{D} \frac{s i n y}{1 + c o s x c o s y} d x d y w i t h$ $D = {[0, \frac{π}{2}]}^{2}$ $3) f i n d t h e v a l u e o f I .$

Question Number 27666 Answers: 0 Comments: 0

let give I_n = ∫_0 ^1 (x^n /(1+x^n ))dx (1) prove that lim_(n−>∝) I_n =0 (2)calculate I_n +I_(n+1) (3) find Σ_(n=1) ^∝ (((−1)^(n−1) )/n) .

$l e t g i v e I_{n} = \int_{0}^{1} \frac{x^{n}}{1 + x^{n}} d x$ $(1) p r o v e t h a t {l i m}_{n - >\propto} I_{n} = 0$ $(2) c a l c u l a t e I_{n} + I_{n + 1}$ $(3) f i n d \sum_{n = 1}^{\propto} \frac{{(- 1)}^{n - 1}}{n} .$

Question Number 28200 Answers: 0 Comments: 1

let give I= ∫_0 ^1 ((ln(1+x))/(1+x^2 ))dx and J=∫∫_([0,1]^2 ) (x/((1+x^2 )(1+xy)))dxdy calculate J by two methods then find the value of I.

$l e t g i v e I = \int_{0}^{1} \frac{l n (1 + x)}{1 + x^{2}} d x a n d J = \int \int_{{[0, 1]}^{2}} \frac{x}{(1 + x^{2}) (1 + x y)} d x d y$ $c a l c u l a t e J b y t w o m e t h o d s t h e n f i n d t h e v a l u e o f I .$

Question Number 27643 Answers: 2 Comments: 1

Question Number 27621 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((arctan(x+(1/x)))/(1+x^2 ))dx .

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

Question Number 27620 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (((−1)^x^2 )/(3+x^2 ))dx .

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{{(- 1)}^{x^{2}}}{3 + x^{2}} d x .$

Question Number 27619 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((cos(2x))/((1+x^2 )^2 ))dx.

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

Question Number 27616 Answers: 0 Comments: 1

find ∫_0 ^1 e^(−2x) ln(1+x)dx .

$f i n d \int_{0}^{1} e^{- 2 x} l n (1 + x) d x .$

Question Number 27615 Answers: 0 Comments: 2

∫x^(5/2) (1−x)^(3/2) dx

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

Question Number 27614 Answers: 0 Comments: 2

∫((cosx)/(2−cosx))dx

$\int \frac{c o s x}{2 - c o s x} d x$

Question Number 27613 Answers: 1 Comments: 1

find the value of ∫_0 ^∞ e^(−[x] −x) dx .

$f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- [x] - x} d x .$

Question Number 27612 Answers: 1 Comments: 0

∫(1/(3+cos^2 x))dx

$\int \frac{1}{3 + {c o s}^{2} x} d x$

Question Number 27611 Answers: 1 Comments: 0

∫(1/(2sin^2 x + 4cos^2 x))dx

$\int \frac{1}{2 \sin^{2} x + 4 \cos^{2} x} d x$

Question Number 27607 Answers: 0 Comments: 0

Question Number 27600 Answers: 0 Comments: 1

find ∫_0 ^π (t/(2+sint)) dt

$f i n d \int_{0}^{π} \frac{t}{2 + s i n t} d t$

Question Number 27597 Answers: 0 Comments: 0

find ∫ ((√(cos(2x)))/(cosx)) dx.

$f i n d \int \frac{\sqrt{c o s (2 x)}}{c o s x} d x .$

Question Number 27596 Answers: 0 Comments: 1

find ∫ ^3 (√( x^2 −x^3 )) dx

$f i n d \int^{3} \sqrt{x^{2} - x^{3}} d x$

Question Number 27595 Answers: 0 Comments: 1

find ∫∫_D xy(√( x^2 +y^2 )) dxdy with D={ (x,y)∈R^2 / x^2 +2y^2 ≤1 ,x≥0 ,y ≥0}

$f i n d \int \int_{D} x y \sqrt{x^{2} + y^{2}} d x d y w i t h$ $D = {(x, y) \in R^{2} / x^{2} + 2 y^{2} ⩽ 1, x ⩾ 0, y ⩾ 0}$

Question Number 27539 Answers: 2 Comments: 1

Question Number 27502 Answers: 0 Comments: 1

find ∫_0 ^(π/2) ((ln(1+xsin^2 t))/(sin^2 t))dt with −1<x<1 .

$f i n d \int_{0}^{\frac{π}{2}} \frac{l n (1 + {x s i n}^{2} t)}{{s i n}^{2} t} d t w i t h - 1 < x < 1 .$

Question Number 27500 Answers: 0 Comments: 2

find ∫∫_Δ (√(4 −x^2 −y^2 )) dxdy with Δ={(x,y) ∈R^2 / x^2 +y^2 ≤2x}

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

Question Number 27496 Answers: 0 Comments: 0

let give f(x)= ∫_0 ^∝ (1/(√t)) e^(−(1+ix)t) dt calculate f^′ (x) prove that ∃λ∈R/(x+i)^2 (f(x))^2 = λ then find ∫_0 ^∝ e^(−t^2 ) dt .

$l e t g i v e f (x) = \int_{0}^{\propto} \frac{1}{\sqrt{t}} e^{- (1 + i x) t} d t$ $c a l c u l a t e f^{^{'}} (x) p r o v e t h a t \exists λ \in R / {(x + i)}^{2} {(f (x))}^{2} = λ$ $t h e n f i n d \int_{0}^{\propto} e^{- t^{2}} d t .$

Question Number 27495 Answers: 0 Comments: 1

find α and β from R /∫_0 ^π (αt^2 +βt)cos(nt)dt= (1/n^2 ) for all number n from N^(∗ ) then find Σ_(n=1) ^∝ (1/n^2 ) .

$f i n d α a n d β f r o m R / \int_{0}^{π} (α t^{2} + β t) c o s (n t) d t = \frac{1}{n^{2}}$ $f o r a l l n u m b e r n f r o m N^{*} t h e n f i n d$ $\sum_{n = 1}^{\propto} \frac{1}{n^{2}} .$

Question Number 27481 Answers: 0 Comments: 1

find the value of ∫_0 ^∝ ((√x)/(e^x −1))dx .

$f i n d t h e v a l u e o f \int_{0}^{\propto} \frac{\sqrt{x}}{e^{x} - 1} d x .$

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