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IntegrationQuestion and Answers: Page 238

Question Number 57746 Answers: 0 Comments: 4

let f(x)=∫_(−∞) ^(+∞) (dt/((t^2 −2xt +1)^2 )) with ∣x∣<1 (x real) 1) determine a explicit form for f(x) 2) find also g(x) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2xt +1)^3 )) 3) calculate ∫_(−∞) ^(+∞) (dt/((t^2 −(√2)t +1)^2 )) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 −(√2)t +1)^3 )) 4) calculate A(θ) =∫_(−∞) ^(+∞) (dt/((t^2 −2cosθ t+1)^2 )) and B(θ) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2cosθ t +1)^3 )) with 0<θ <(π/2) .

$l e t f (x) = \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - 2 x t + 1)}^{2}} w i t h ∣ x ∣< 1 (x r e a l)$ $1) d e t e r m i n e a e x p l i c i t f o r m f o r f (x)$ $2) f i n d a l s o g (x) = \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - 2 x t + 1)}^{3}}$ $3) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - \sqrt{2} t + 1)}^{2}} a n d \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - \sqrt{2} t + 1)}^{3}}$ $4) c a l c u l a t e A (θ) = \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - 2 c o s θ t + 1)}^{2}} a n d$ $B (θ) = \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - 2 c o s θ t + 1)}^{3}} w i t h 0 < θ < \frac{π}{2} .$

Question Number 57668 Answers: 0 Comments: 3

let V_n = ∫_0 ^(1+(1/n)) ((x+1)/(√(2x^2 +3))) dx 1) calculate lim_(n→+∞) V_n 2) find nature of the serie Σ V_n

$l e t V_{n} = \int_{0}^{1 + \frac{1}{n}} \frac{x + 1}{\sqrt{2 x^{2} + 3}} d x$ $1) c a l c u l a t e {l i m}_{n \to + \infty} V_{n}$ $2) f i n d n a t u r e o f t h e s e r i e Σ V_{n}$

Question Number 57667 Answers: 0 Comments: 3

calculate U_n =∫_(π/n) ^((2π)/n) (dx/(2 +sinx)) 1) calculate U_n and lim_(n→+∞) nU_n 2) find nature of Σ U_n

$c a l c u l a t e U_{n} = \int_{\frac{π}{n}}^{\frac{2 π}{n}} \frac{d x}{2 + s i n x}$ $1) c a l c u l a t e U_{n} a n d {l i m}_{n \to + \infty} {n U}_{n}$ $2) f i n d n a t u r e o f Σ U_{n}$

Question Number 57666 Answers: 0 Comments: 3

1) calculate f(θ) =∫_0 ^1 (√(t^2 +2sinθt +1))dt with 0≤θ≤(π/2) 2) calculate g(t) =∫_0 ^1 (√(t^2 +2(sinθ)t +1))dθ 3) find also h(θ) =∫_0 ^1 (t/(√(t^2 +2(sinθ)t +1)))dt

$1) c a l c u l a t e f (θ) = \int_{0}^{1} \sqrt{t^{2} + 2 s i n θ t + 1} d t w i t h 0 ⩽ θ ⩽ \frac{π}{2}$ $2) c a l c u l a t e g (t) = \int_{0}^{1} \sqrt{t^{2} + 2 (s i n θ) t + 1} d θ$ $3) f i n d a l s o h (θ) = \int_{0}^{1} \frac{t}{\sqrt{t^{2} + 2 (s i n θ) t + 1}} d t$

Question Number 57665 Answers: 0 Comments: 4

let f(a) =∫_(π/4) ^(π/3) (√(a+tan^2 x))dx with a>0 1) find a explicit form of f(a) 2) find also g(a) =∫_(π/4) ^(π/3) (dx/(√(a+tan^2 x))) 3) find the values of ∫_(π/4) ^(π/3) (√(2+tan^2 x))dx and ∫_(π/4) ^(π/3) (dx/(√(3+tan^2 x)))

$l e t f (a) = \int_{\frac{π}{4}}^{\frac{π}{3}} \sqrt{a + {t a n}^{2} x} d x w i t h a > 0$ $1) f i n d a e x p l i c i t f o r m o f f (a)$ $2) f i n d a l s o g (a) = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{\sqrt{a + {t a n}^{2} x}}$ $3) f i n d t h e v a l u e s o f \int_{\frac{π}{4}}^{\frac{π}{3}} \sqrt{2 + {t a n}^{2} x} d x a n d \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{\sqrt{3 + {t a n}^{2} x}}$

Question Number 57653 Answers: 0 Comments: 5

is it possible to find the exact value of I? I=∫_0 ^π sin (sin x) dx

$is it possible to find the exact value of I ?$ $I = \underset{0}{\int^{π}} \sin (\sin x) d x$

Question Number 57490 Answers: 1 Comments: 2

1)findF(a)= ∫_0 ^∞ ((cos(ln(2+x^2 )))/(a^2 +x^2 ))dx witha>0 2) find the value of ∫_0 ^∞ ((cos(ln(2+x^2 )))/(4+x^2 ))dx.

$1) f i n d F (a) = \int_{0}^{\infty} \frac{c o s (l n (2 + x^{2}))}{a^{2} + x^{2}} d x w i t h a > 0$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{c o s (l n (2 + x^{2}))}{4 + x^{2}} d x .$

Question Number 57487 Answers: 0 Comments: 1

calculate lim_(x→1) ∫_x ^x^2 ((arctan(t))/(sint))dt .

$c a l c u l a t e {l i m}_{x \to 1} \int_{x}^{x^{2}} \frac{a r c t a n (t)}{s i n t} d t .$

Question Number 57423 Answers: 0 Comments: 0

let A_n =∫_0 ^∞ (dt/((e^t +e^(−t) )^n )) calculate A_n interms of n

$l e t A_{n} = \int_{0}^{\infty} \frac{d t}{{(e^{t} + \overset{- t}{e})}^{n}}$ $c a l c u l a t e A_{n} i n t e r m s o f n$

Question Number 57421 Answers: 1 Comments: 0

calculate ∫_(−1) ^1 (((x^4 +x^2 +1)^2 +e^x )/(e^x +1))dx

$c a l c u l a t e \int_{- 1}^{1} \frac{{(x^{4} + x^{2} + 1)}^{2} + e^{x}}{e^{x} + 1} d x$

Question Number 57420 Answers: 0 Comments: 1

let J(x)=∫_0 ^x (t^2 /((√(t+1)) +(√(t+4))))dt find a explicit form of J(x)

$l e t J (x) = \int_{0}^{x} \frac{t^{2}}{\sqrt{t + 1} + \sqrt{t + 4}} d t$ $f i n d a e x p l i c i t f o r m o f J (x)$

Question Number 57419 Answers: 0 Comments: 1

find ∫_0 ^1 (x+1) ln(x+(√(1+x^2 )))dx

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

Question Number 57418 Answers: 0 Comments: 1

calculate ∫_(−1) ^4 ((∣x−1∣+∣x−2∣)/(∣x^2 −9∣ +x^2 +16))dx

$c a l c u l a t e \int_{- 1}^{4} \frac{∣ x - 1 ∣ + ∣ x - 2 ∣}{∣ x^{2} - 9 ∣ + x^{2} + 16} d x$

Question Number 57417 Answers: 0 Comments: 2

let F(x) =∫_0 ^x ((1+sint)/(2+cost))dt 1) find a explicite form of f(x) 2) calculate ∫_0 ^π ((1+sint)/(2+cost))dt

$l e t F (x) = \int_{0}^{x} \frac{1 + s i n t}{2 + c o s t} d t$ $1) f i n d a e x p l i c i t e f o r m o f f (x)$ $2) c a l c u l a t e \int_{0}^{π} \frac{1 + s i n t}{2 + c o s t} d t$

Question Number 57388 Answers: 0 Comments: 0

Given f(x) = f(x + 2016), ∀x ∈ R If ∫_0 ^3 f(x) = 30, then ∫_3 ^5 f(x + 2016) = ...

$Given f (x) = f (x + 2016), \forall x \in R$ $If \underset{0}{\int^{3}} f (x) = 30, then \underset{3}{\int^{5}} f (x + 2016) = . . .$

Question Number 57385 Answers: 1 Comments: 1

Question Number 57325 Answers: 0 Comments: 1

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

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

Question Number 57324 Answers: 0 Comments: 0

we want to find the vslue of I =∫_0 ^1 ((ln(1+x))/(1+x^2 )) dx let A=∫∫_W (x/((1+x^2 )(1+xy)))dxdy with W=[0,1]^2 calculate A by two method and conclude the value of I .

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

Question Number 57323 Answers: 0 Comments: 1

calculate ∫∫_D ((x+y)/(3+(√(x^2 +y^2 ))))dxdy with D={(x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0 ,y≥0}

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

Question Number 57321 Answers: 1 Comments: 1

calculate ∫∫_D (x−y)(√(x^2 +y^2 ))dxdy with D ={ (x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0}

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

Question Number 57320 Answers: 1 Comments: 1

calculate ∫∫_D xy e^(−x^2 −y^2 ) dxdy with D={(x,y)∈R^2 / 0≤x≤2 and 1≤y≤3}

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

Question Number 57319 Answers: 1 Comments: 1

calculate ∫∫_D e^(x−y) dxdy with D={(x,y)∈R^2 /∣x∣<1 and 0≤y≤1}

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

Question Number 57228 Answers: 0 Comments: 1

find f(x) =∫_1 ^2 ((ln(1+xt))/t^2 ) dt with x>0

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

Question Number 57227 Answers: 0 Comments: 0

let f(α)=∫_0 ^1 ((arctan(αx))/(1+αx^2 )) dx with α real 1) find f(α) interms of α 2) find the values of ∫_0 ^1 ((arctan(2x))/(1+2x^2 )) dx and ∫_0 ^1 ((arctan(4x))/(1+4x^2 ))dx

$l e t f (α) = \int_{0}^{1} \frac{a r c t a n (α x)}{1 + α x^{2}} d x w i t h α r e a l$ $1) f i n d f (α) i n t e r m s o f α$ $2) f i n d t h e v a l u e s o f \int_{0}^{1} \frac{a r c t a n (2 x)}{1 + 2 x^{2}} d x a n d \int_{0}^{1} \frac{a r c t a n (4 x)}{1 + 4 x^{2}} d x$

Question Number 57226 Answers: 0 Comments: 0

calculate A_n =∫_0 ^1 x^n (√((1−x)/(1+x)))dx with n integr natural

$c a l c u l a t e A_{n} = \int_{0}^{1} x^{n} \sqrt{\frac{1 - x}{1 + x}} d x w i t h n i n t e g r n a t u r a l$

Question Number 57224 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((3t^2 −5t +1)/((t+1)(t+2)(2t+3)))dt

$f i n d t h e v a l u e o f \int_{0}^{1} \frac{3 t^{2} - 5 t + 1}{(t + 1) (t + 2) (2 t + 3)} d t$

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