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IntegrationQuestion and Answers: Page 305

Question Number 31102 Answers: 0 Comments: 2

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

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

Question Number 31101 Answers: 0 Comments: 0

let give f(x)= ∫_0 ^x t^2 e^(−2t^2 ) sin(2(x−t))dt calculate f^(′′) +4f then finf f(x).

$l e t g i v e f (x) = \int_{0}^{x} t^{2} e^{- 2 t^{2}} s i n (2 (x - t)) d t c a l c u l a t e$ $f^{^{″}} + 4 f t h e n f i n f f (x) .$

Question Number 31100 Answers: 0 Comments: 1

find ∫_0 ^∞ ((cosx −cos(3x))/x) e^(−2x) dx.

$f i n d \int_{0}^{\infty} \frac{c o s x - c o s (3 x)}{x} e^{- 2 x} d x .$

Question Number 31098 Answers: 0 Comments: 2

find the value of ∫_1 ^∞ ((arctan(x+1) −arctanx)/x^2 )dx.

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

Question Number 31097 Answers: 0 Comments: 1

calculate interms of a and b the integral ∫_0 ^∞ ((arctan(bt) −arctan(at))/t)dt with a and b>0.

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

Question Number 31096 Answers: 0 Comments: 1

find ∫_0 ^π (dx/((a+bcosx)^2 )) with a>b>0 then give the value of ∫_0 ^π (dx/((2+cosx)^2 ))

$f i n d \int_{0}^{π} \frac{d x}{{(a + b c o s x)}^{2}} w i t h a > b > 0 t h e n g i v e t h e$ $v a l u e o f \int_{0}^{π} \frac{d x}{{(2 + c o s x)}^{2}}$

Question Number 31095 Answers: 0 Comments: 1

find I_n (x)= ∫_0 ^∞ t^n e^(−xt) dt x>0 n∈ N.

$f i n d I_{n} (x) = \int_{0}^{\infty} t^{n} e^{- x t} d t x > 0 n \in N .$

Question Number 31094 Answers: 0 Comments: 0

m and n integrs and y≥0 find ∫_0 ^y x^m (y−x)^n dx

$m a n d n i n t e g r s a n d y ⩾ 0 f i n d \int_{0}^{y} x^{m} {(y - x)}^{n} d x$

Question Number 31093 Answers: 0 Comments: 1

calculate ∫_0 ^∞ e^(−x^2 ) cos(2xy)dx.

$c a l c u l a t e \int_{0}^{\infty} e^{- x^{2}} c o s (2 x y) d x .$

Question Number 31092 Answers: 0 Comments: 1

find ∫_0 ^∞ ((ln(1+4x^2 ))/(1+2x^2 ))dx .

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

Question Number 31091 Answers: 0 Comments: 1

let −1<t<1 find f(t)= ∫_0 ^π ((ln(1+tcosx))/(cosx))dx

$l e t - 1 < t < 1 f i n d f (t) = \int_{0}^{π} \frac{l n (1 + t c o s x)}{c o s x} d x$

Question Number 31090 Answers: 0 Comments: 1

find ∫∫_(1≤x^2 +y^2 ≤4 and y≥0) ((dxdy)/(√(x^2 +y^2 ))) .

$f i n d \int \int_{1 ⩽ x^{2} + y^{2} ⩽ 4 a n d y ⩾ 0} \frac{d x d y}{\sqrt{x^{2} + y^{2}}} .$

Question Number 31089 Answers: 0 Comments: 0

find ∫_0 ^1 dy ∫_y^2 ^y ((ydx)/(x(√(x^2 +y^2 )))) .

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

Question Number 31088 Answers: 0 Comments: 0

find ∫_0 ^1 dx ∫_0 ^(1−x) e^((y−x)/(y+x)) dy.

$f i n d \int_{0}^{1} d x \int_{0}^{1 - x} e^{\frac{y - x}{y + x}} d y .$

Question Number 31087 Answers: 0 Comments: 0

find ∫∫∫_(x^2 +y^2 +z^2 <4) (x^2 +y^2 +z^2 )dxdydz.

$f i n d \int \int \int_{x^{2} + y^{2} + z^{2} < 4} (x^{2} + y^{2} + z^{2}) d x d y d z .$

Question Number 31086 Answers: 0 Comments: 0

find ∫∫_D (x^4 −y^4 )dxdy with D= {(x,y)∈R^2 / 1<x^2 −y^2 <2 ,1<xy<2 ,x>0,y>0}

$f i n d \int \int_{D} (x^{4} - y^{4}) d x d y w i t h$ $D = {(x, y) \in R^{2} / 1 < x^{2} - y^{2} < 2, 1 < x y < 2, x > 0, y > 0}$

Question Number 31084 Answers: 0 Comments: 1

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

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

Question Number 31083 Answers: 0 Comments: 1

calculate by two methods ∫_0 ^1 ∫_0 ^(π/2) ((dx dt)/(1+x^2 tan^2 t)) then find the value of ∫_0 ^(π/2) t cotant dt .

$c a l c u l a t e b y t w o m e t h o d s \int_{0}^{1} \int_{0}^{\frac{π}{2}} \frac{d x d t}{1 + x^{2} {t a n}^{2} t}$ $t h e n f i n d t h e v a l u e o f \int_{0}^{\frac{π}{2}} t c o t a n t d t .$

Question Number 31082 Answers: 0 Comments: 0

calculate by two methods ∫_0 ^∞ ∫_0 ^∞ ((dxdy)/((1+y)(1+x^2 y))) then find the value of ∫_0 ^∞ ((lnx)/(1−x^2 ))dx.

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

Question Number 31081 Answers: 0 Comments: 0

find ∫_0 ^∞ dx ∫_x ^(+∞) e^(−y^2 dy) .

$f i n d \int_{0}^{\infty} d x \int_{x}^{+ \infty} e^{- y^{2} d y} .$

Question Number 31080 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−px) dx ∫_0 ^a ((cos(xt))/(√(a^2 −t^2 )))dt with a>0 ,p>0

$f i n d \int_{0}^{\infty} e^{- p x} d x \int_{0}^{a} \frac{c o s (x t)}{\sqrt{a^{2} - t^{2}}} d t w i t h a > 0, p > 0$

Question Number 31079 Answers: 0 Comments: 0

calculate ∫∫_(0≤x≤1 and 0≤y≤2) x^2 y e^(xy) dxdxy.

$c a l c u l a t e \int \int_{0 ⩽ x ⩽ 1 a n d 0 ⩽ y ⩽ 2} x^{2} y e^{x y} d x d x y .$

Question Number 31078 Answers: 0 Comments: 0

find ∫∫_(0≤x≤3 and x≤y≤4x−x^2 ) (x^2 +2y)dxdy.

$f i n d \int \int_{0 ⩽ x ⩽ 3 a n d x ⩽ y ⩽ 4 x - x^{2}} (x^{2} + 2 y) d x d y .$

Question Number 31077 Answers: 0 Comments: 1

calculate ∫∫_(0<x<1and 0<y<x^2 ) (y/(√(x^2 +y^2 )))dxdy.

$c a l c u l a t e \int \int_{0 < x < 1 a n d 0 < y < x^{2}} \frac{y}{\sqrt{x^{2} + y^{2}}} d x d y .$

Question Number 31076 Answers: 0 Comments: 1

find ∫_0 ^((√2)/2) (dx/((2x^2 +1)(√(1+x^2 )))) .

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

Question Number 31075 Answers: 0 Comments: 0

find ∫_0 ^π ((sinθ)/(cos^2 θ +2 sin^2 θ)) dθ .

$f i n d \int_{0}^{π} \frac{s i n θ}{{c o s}^{2} θ + 2 {s i n}^{2} θ} d θ .$

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