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IntegrationQuestion and Answers: Page 232

Question Number 61855 Answers: 1 Comments: 0

∫_(0 ) ^1 ((3x^3 −x^2 +2x−4)/(√(x^2 −3x+2))) dx

$\int_{0}^{1} \frac{3 x^{3} - x^{2} + 2 x - 4}{\sqrt{x^{2} - 3 x + 2}} d x$

Question Number 61809 Answers: 1 Comments: 1

Question Number 61803 Answers: 0 Comments: 3

find ∫_0 ^∞ (x^2 /(e^x^2 −1))dx

$f i n d \int_{0}^{\infty} \frac{x^{2}}{e^{x^{2}} - 1} d x$

Question Number 61801 Answers: 0 Comments: 3

∫_(2π) ^(4π) (√(1−cos(x))) dx

$\underset{2 π}{\int^{4 π}} \sqrt{1 - c o s (x)} d x$

Question Number 61785 Answers: 1 Comments: 0

∫_0 ^(√(3−x^2 )) ((xy(4−x^2 −y^2 )(√(4−x^2 −y^2 ))−xy)/3) dy

$\underset{0}{\int^{\sqrt{3 - x^{2}}}} \frac{x y (4 - x^{2} - y^{2}) \sqrt{4 - x^{2} - y^{2}} - x y}{3} d y$

Question Number 61748 Answers: 0 Comments: 1

find ∫ (dx/(sin(2x)+tan(x)))dx

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

Question Number 61750 Answers: 0 Comments: 0

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

$f i n d \int \frac{x^{2} - \sqrt{x - 1}}{2 \sqrt{x^{2} + 3}} d x$

Question Number 61749 Answers: 0 Comments: 0

find ∫ (dx/(cos(2x)+tan(x)))

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

Question Number 61735 Answers: 0 Comments: 0

∫_(−1) ^1 (((sin(x))/(sinh^(−1) (x))))(((sin^(−1) (x))/(sinh(x)))) dx =?

$\underset{- 1}{\int^{1}} (\frac{s i n (x)}{{s i n h}^{- 1} (x)}) (\frac{{s i n}^{- 1} (x)}{s i n h (x)}) d x = ?$

Question Number 61721 Answers: 0 Comments: 0

calculate A =∫_0 ^∞ cos(x^n )dx and B =∫_0 ^∞ sin(x^n )dx with n≥2 (n integr natural)

$c a l c u l a t e A = \int_{0}^{\infty} c o s (x^{n}) d x$ $a n d B = \int_{0}^{\infty} s i n (x^{n}) d x w i t h$ $n ⩾ 2 (n i n t e g r n a t u r a l)$

Question Number 61719 Answers: 0 Comments: 1

∫(dx/(2+sin(x)))

$\int \frac{d x}{2 + s i n (x)}$

Question Number 61662 Answers: 0 Comments: 1

calculate ∫_(−(π/4)) ^(π/4) ((cosx)/(e^(1/x) +1)) dx

$c a l c u l a t e \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{c o s x}{e^{\frac{1}{x}} + 1} d x$

Question Number 61661 Answers: 0 Comments: 1

1) calculate ∫∫_R^+^2 ((dxdy)/((1+x^2 )(1+y^2 ))) 2) find the value of ∫_0 ^∞ ((ln(x))/(x^2 −1)) dx .

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

Question Number 61660 Answers: 0 Comments: 1

let U_n = ∫_0 ^∞ (dt/((1+t^3 )^n )) dt (n≥1) 1) calculate (U_(n+1) /U_n ) 2) study the serie Σln((U_(n+1) /U_n )) and prove that lim_(n→+∞) U_n =0

$l e t U_{n} = \int_{0}^{\infty} \frac{d t}{{(1 + t^{3})}^{n}} d t (n ⩾ 1)$ $1) c a l c u l a t e \frac{U_{n + 1}}{U_{n}}$ $2) s t u d y t h e s e r i e Σ l n (\frac{U_{n + 1}}{U_{n}}) a n d p r o v e t h a t {l i m}_{n \to + \infty} U_{n} = 0$

Question Number 61674 Answers: 1 Comments: 4

a.∫_( 0) ^( (𝛑/4)) (√(1+tgx)) dx=? b.∫_( 0) ^( 1) (√(1+lnx)) dx=?

$a . \underset{0}{\int^{\frac{π}{4}}} \sqrt{1 + tgx} dx = ?$ $b . \underset{0}{\int^{1}} \sqrt{1 + lnx} dx = ?$

Question Number 61654 Answers: 0 Comments: 0

∫_0 ^∞ e^(−e^x ) ln(x) dx = 0.27634

$\int_{0}^{\infty} e^{- e^{x}} l n (x) d x = 0.27634$

Question Number 61645 Answers: 0 Comments: 1

calculate ∫∫_D ∫(√(x^2 +y^2 +z^2 ))dxdydz with D ={(x,y,z) / 0≤x≤1 ,1≤y≤2 , 2≤z≤3 }

$c a l c u l a t e \int \int_{D} \int \sqrt{x^{2} + y^{2} + z^{2}} d x d y d z$ $w i t h D = {(x, y, z) / 0 ⩽ x ⩽ 1, 1 ⩽ y ⩽ 2, 2 ⩽ z ⩽ 3}$

Question Number 61648 Answers: 0 Comments: 1

calculate ∫∫_W (x^2 −2y^2 )(√(x^2 +y^2 +3))dxdy with W ={ (x,y) ∈ R^2 / 1≤x ≤(√3) and x^2 +y^2 −2y ≤ 2 }

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

Question Number 61667 Answers: 1 Comments: 0

∫(√(tan(x))) dx

$\int \sqrt{t a n (x)} d x$

Question Number 61601 Answers: 1 Comments: 1

calvulate ∫∫_w (x^2 −y^2 )e^(−x−y) dxdy with W={(x,y)∈R^2 /0≤x≤1 and 1≤y≤3}

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

Question Number 61566 Answers: 1 Comments: 0

∫_2 ^4 ((√(ln(9−(6−x)))/((√(ln(9−x))) + (√(ln(3−x))))) dx

$\int_{2}^{4} \frac{\sqrt{l n (9 - (6 - x)}}{\sqrt{l n (9 - x)} + \sqrt{l n (3 - x)}} d x$

Question Number 61545 Answers: 0 Comments: 0

Question Number 61535 Answers: 0 Comments: 0

calculate ∫_0 ^(π/2) ((ln(1+cosx))/(cosx)) dx

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{l n (1 + c o s x)}{c o s x} d x$

Question Number 61534 Answers: 0 Comments: 0

calculate f(a) =∫∫_W (x+ay)e^(−x) e^(−ay) dxdy with W_a ={(x,y)∈R^2 /x≥0 ,y≥0 , x+ay ≤1 } a>0

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

Question Number 61533 Answers: 0 Comments: 2

∫∫_([0,1]^2 ) ((x−y)/((x^2 +3y^(2 ) +1)^2 )) dxdy

$\int \int_{{[0, 1]}^{2}} \frac{x - y}{{(x^{2} + 3 y^{2} + 1)}^{2}} d x d y$

Question Number 61530 Answers: 0 Comments: 5

let U_n =∫_0 ^∞ (x^(−2n) /(1+x^4 )) dx with n integr natural and n≥1 1) calculate U_n interms of n 2) find lim_(n→+∞) n^2 U_n 3) study the serie Σ U_n

$l e t U_{n} = \int_{0}^{\infty} \frac{x^{- 2 n}}{1 + x^{4}} d x w i t h n i n t e g r n a t u r a l a n d n ⩾ 1$ $1) c a l c u l a t e U_{n} i n t e r m s o f n$ $2) f i n d {l i m}_{n \to + \infty} n^{2} U_{n}$ $3) s t u d y t h e s e r i e Σ U_{n}$

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