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IntegrationQuestion and Answers: Page 313

Question Number 27999 Answers: 0 Comments: 1

find I_(n,m) = ∫_0 ^1 x^n (1−x)^m dx with (n,m)∈N^★^2 and calculate Σ_(n=0) ^∝ I_(n,m) .

$f i n d I_{n, m} = \int_{0}^{1} x^{n} {(1 - x)}^{m} d x w i t h$ $(n, m) \in N^{★^{2}} a n d c a l c u l a t e \sum_{n = 0}^{\propto} I_{n, m} .$

Question Number 27974 Answers: 0 Comments: 1

let put f(t)=∫_0 ^∞ ((e^(−ax) − e^(−bx) )/x^2 ) e^(−tx^2 ) dx with t≥0 and a>0 and b>0 find a integral form of f(t).

$l e t p u t f (t) = \int_{0}^{\infty} \frac{e^{- a x} - e^{- b x}}{x^{2}} e^{- {t x}^{2}} d x$ $w i t h t ⩾ 0 a n d a > 0 a n d b > 0$ $f i n d a i n t e g r a l f o r m o f f (t) .$

Question Number 27951 Answers: 0 Comments: 0

Use the trapezoidal rule with 5 ordinates to evaluate ∫_( 0) ^( 0.8) e^x^2 dx

$Use the trapezoidal rule with 5 ordinates to evaluate \int_{0}^{0.8} e^{x^{2}} dx$

Question Number 27950 Answers: 0 Comments: 0

Find by the trapezoidal rule the approximate value of ∫_( 0) ^( 1) (dx/(1 + x^2 )). Use ordinates spaced at equal interval of width h = 0.1

$Find by the trapezoidal rule the approximate value of \int_{0}^{1} \frac{dx}{1 + x^{2}} . Use$ $ordinates spaced at equal interval of width h = 0.1$

Question Number 27878 Answers: 0 Comments: 1

Question Number 27853 Answers: 0 Comments: 1

Question Number 27828 Answers: 1 Comments: 2

find the value of ∫_0 ^(π/2) (√(tanx))dx .

$f i n d t h e v a l u e o f \int_{0}^{\frac{π}{2}} \sqrt{t a n x} d x .$

Question Number 27815 Answers: 1 Comments: 0

∫((cos x−cos 2x)/(1−cos x))dx

$\int \frac{\cos x - \cos 2 x}{1 - \cos x} dx$

Question Number 27805 Answers: 0 Comments: 1

find ∫_1 ^∝ ((arctan(αx))/x^2 ) .

$f i n d \int_{1}^{\propto} \frac{a r c t a n (α x)}{x^{2}} .$

Question Number 27804 Answers: 0 Comments: 1

calculate ∫_0 ^∝ ((e^(−ax) − e^(−bx) )/x^2 )dx with a>0 b>o

$c a l c u l a t e \int_{0}^{\propto} \frac{e^{- a x} - e^{- b x}}{x^{2}} d x w i t h a > 0$ $b > o$

Question Number 27803 Answers: 0 Comments: 0

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

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

Question Number 27802 Answers: 0 Comments: 1

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

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

Question Number 27797 Answers: 0 Comments: 1

find ∫ (√(2+tan^2 t)) dt.

$f i n d \int \sqrt{2 + {t a n}^{2} t} d t .$

Question Number 27796 Answers: 0 Comments: 0

find ∫ (x^2 /((cosx +x sinx)^2 )) .

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

Question Number 27794 Answers: 0 Comments: 0

let give I(x)= ∫_0 ^π ln (1−2x cost +x^2 )dt by using the polynomial p(x)= (z+x)^(2n) −1 find the value of I(x).

$l e t g i v e I (x) = \int_{0}^{π} l n (1 - 2 x c o s t + x^{2}) d t b y u s i n g t h e$ $p o l y n o m i a l p (x) = {(z + x)}^{2 n} - 1 f i n d t h e v a l u e o f I (x) .$

Question Number 27788 Answers: 0 Comments: 0

find the value of A_n = ∫_0 ^π ((sin(nt))/(sint))dt with n∈N^∗ .

$f i n d t h e v a l u e o f A_{n} = \int_{0}^{π} \frac{s i n (n t)}{s i n t} d t w i t h n \in N^{*} .$

Question Number 27781 Answers: 0 Comments: 1

find the value of F(x)=∫_0 ^(π/2) ((ln(1+x sin^2 t))/(sin^2 t)) dt knowing that −1<x<1 .

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

Question Number 27764 Answers: 1 Comments: 0

∫(√(tan x))dx

$\int \sqrt{\tan x} d x$

Question Number 27693 Answers: 1 Comments: 1

1) calculate ∫∫_(]0,1]×]0,(π/2)]) ((dxdy)/(1+(xtany)^2 )) 2) find the value of ∫_0 ^(π/2) (t/(tant))dt .

$1) c a l c u l a t e \int \int_{] 0, 1] \times] 0, \frac{π}{2}]} \frac{d x d y}{1 + {(x t a n y)}^{2}}$ $2) f i n d t h e v a l u e o f \int_{0}^{\frac{π}{2}} \frac{t}{t a n t} d t .$

Question Number 27692 Answers: 0 Comments: 1

find by two ways the value of ∫∫_([0,1]) x^y dxdxy then calculate ∫_0 ^1 ((t−1)/(lnt))dt .

$f i n d b y t w o w a y s t h e v a l u e o f \int \int_{[0, 1]} x^{y} d x d x y t h e n$ $c a l c u l a t e \int_{0}^{1} \frac{t - 1}{l n t} d t .$

Question Number 27691 Answers: 0 Comments: 1

let give A=∫∫_(0≤y≤x≤1) ((dxdxy)/((1+x^2 )(1+y^2 ))) and B= ∫_0 ^(π/4) ((ln(2cos^2 θ))/(2cos(2θ)))dθ calculate A and prove that B=A.

$l e t g i v e A = \int \int_{0 ⩽ y ⩽ x ⩽ 1} \frac{d x d x y}{(1 + x^{2}) (1 + y^{2})} a n d$ $B = \int_{0}^{\frac{π}{4}} \frac{l n (2 {c o s}^{2} θ)}{2 c o s (2 θ)} d θ c a l c u l a t e A a n d p r o v e t h a t B = A .$

Question Number 27690 Answers: 0 Comments: 1

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

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

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) .

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

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