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IntegrationQuestion and Answers: Page 287

Question Number 35677 Answers: 0 Comments: 2

find F(x)=∫_0 ^x e^(−2t) cos(t+(π/4))dx.

$f i n d F (x) = \int_{0}^{x} e^{- 2 t} c o s (t + \frac{π}{4}) d x .$

Question Number 35676 Answers: 0 Comments: 1

find f(x)=∫_0 ^x ch^4 t dt

$f i n d f (x) = \int_{0}^{x} {c h}^{4} t d t$

Question Number 35675 Answers: 0 Comments: 1

calculate ∫_1 ^3 (x/(e^x −1))dx ..

$c a l c u l a t e \int_{1}^{3} \frac{x}{e^{x} - 1} d x . .$

Question Number 35635 Answers: 1 Comments: 1

Question Number 35992 Answers: 0 Comments: 1

let f(x)= ((sin(2x))/x) χ_(]−a,a[) (x) with a>0 calculate the fourier trsnsform of f .

$l e t f (x) = \frac{s i n (2 x)}{x} χ_{] - a, a [} (x) w i t h a > 0$ $c a l c u l a t e t h e f o u r i e r t r s n s f o r m o f f .$

Question Number 35632 Answers: 0 Comments: 2

let ϕ(x)= (1/(√(a^2 −x^2 ))) if ∣x∣<a and ϕ(x)=0 if ∣x∣≥a find the fourier transform of ϕ .

$l e t φ (x) = \frac{1}{\sqrt{a^{2} - x^{2}}} i f ∣ x ∣< a a n d φ (x) = 0 i f ∣ x ∣⩾ a$ $f i n d t h e f o u r i e r t r a n s f o r m o f φ .$

Question Number 35631 Answers: 0 Comments: 0

let U_n = ∫_0 ^∞ e^(−(t/n)) arctan(t)dt find a equivalent of U_n (n→+∞)

$l e t U_{n} = \int_{0}^{\infty} e^{- \frac{t}{n}} a r c t a n (t) d t$ $f i n d a e q u i v a l e n t o f U_{n} (n \to + \infty)$

Question Number 35630 Answers: 0 Comments: 5

1) find the value of f(x)=∫_0 ^∞ ((1−cos(xt))/t^2 ) e^(−t) dt 2) calculate ∫_0 ^∞ ((1−cos(t))/t^2 ) e^(−t) dt .

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

Question Number 35629 Answers: 0 Comments: 0

let f(x,y) = ∫_x ^y ((ln(t)ln(1−t))/t)dt with 0<x<y<1 give f(x,y) at form of serie .

$l e t f (x, y) = \int_{x}^{y} \frac{l n (t) l n (1 - t)}{t} d t w i t h 0 < x < y < 1$ $g i v e f (x, y) a t f o r m o f s e r i e .$

Question Number 35628 Answers: 0 Comments: 1

find the value of I =∫_0 ^1 ((ln(t)ln(1−t))/t)dt

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

Question Number 35627 Answers: 0 Comments: 0

study the convergence of I =∫_0 ^∞ (dx/((1+x^2 ∣sinx∣)^(3/2) ))

$s t u d y t h e c o n v e r g e n c e o f$ $I = \int_{0}^{\infty} \frac{d x}{{(1 + x^{2} ∣ s i n x ∣)}^{\frac{3}{2}}}$

Question Number 35625 Answers: 0 Comments: 0

find lim_(ξ→0) ∫_0 ^(π/2) (dx/(√( sin^2 x +ξ cos^2 x)))

$f i n d {l i m}_{ξ \to 0} \int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{{s i n}^{2} x + ξ {c o s}^{2} x}}$

Question Number 35620 Answers: 0 Comments: 1

let f(x) =e^(−x) sinx odd 2π periodic developp f at fourier serie .

$l e t f (x) = e^{- x} s i n x o d d 2 π p e r i o d i c$ $d e v e l o p p f a t f o u r i e r s e r i e .$

Question Number 35619 Answers: 0 Comments: 2

let f(x) = x∣x∣ odd 2π periodic developp f at fourier serie .

$l e t f (x) = x ∣ x ∣ o d d 2 π p e r i o d i c$ $d e v e l o p p f a t f o u r i e r s e r i e .$

Question Number 35618 Answers: 0 Comments: 1

integrate the e.d. y′ +e^(−2x) y = (2x+1)cosx

$i n t e g r a t e t h e e . d . y^{'} + e^{- 2 x} y = (2 x + 1) c o s x$

Question Number 35617 Answers: 0 Comments: 0

integrate the e.d . y^(′′) +(x−1)y = e^(−x) sinx with y(0) =1

$i n t e g r a t e t h e e . d . y^{^{″}} + (x - 1) y = e^{- x} s i n x$ $w i t h y (0) = 1$

Question Number 35616 Answers: 0 Comments: 0

integrate the d.e y^(′′) −2y^′ +y = x^2 ch(x)

$i n t e g r a t e t h e d . e y^{^{″}} - 2 y^{^{'}} + y = x^{2} c h (x)$

Question Number 35615 Answers: 0 Comments: 0

let S_n = Σ_(k=0) ^n (1/(3k+1)) calculate S_n interms of H_n with H_n =Σ_(k=1) ^n (1/k)

$l e t S_{n} = \sum_{k = 0}^{n} \frac{1}{3 k + 1}$ $c a l c u l a t e S_{n} i n t e r m s o f H_{n} w i t h H_{n} = \sum_{k = 1}^{n} \frac{1}{k}$

Question Number 35613 Answers: 0 Comments: 0

find I_(a,b) = ∫_(−∞) ^(+∞) (e^x /((1+a e^x )(1+be^x )))dx ..

$f i n d I_{a, b} = \int_{- \infty}^{+ \infty} \frac{e^{x}}{(1 + a e^{x}) (1 + {b e}^{x})} d x . .$

Question Number 35612 Answers: 0 Comments: 0

calculate I =∫_0 ^∞ (((1+t)^(−(1/4)) −(1+t)^(−(3/4)) )/t)dt

$c a l c u l a t e I = \int_{0}^{\infty} \frac{{(1 + t)}^{- \frac{1}{4}} - {(1 + t)}^{- \frac{3}{4}}}{t} d t$

Question Number 35611 Answers: 0 Comments: 0

let h(t) = e^(t−e^t ) and for n≥0 we put h_n (t) =nh(nt) calculate ∫_(−∞) ^(+∞) h_n (t)dt .

$l e t h (t) = e^{t - e^{t}} a n d f o r n ⩾ 0 w e p u t$ $h_{n} (t) = n h (n t)$ $c a l c u l a t e \int_{- \infty}^{+ \infty} h_{n} (t) d t .$

Question Number 35610 Answers: 0 Comments: 0

let give x∈]0,2π[ and a ∈R,b∈ R prove that ((π−x)/2) = arctan(((sinx)/(1−cosx))) 2) prove that ∣arctan(a)−arctan(b)∣≤∣a−b∣ 3)letθ ∈]0,(π/2)[ , x ∈[θ,2π−θ] , r∈[0,1[ prove that ∣ϕ(x,r) −((π−x)/2)∣≤ ((1−r)/((1−cosθ)^2 ))

$l e t g i v e x \in] 0, 2 π [a n d a \in R, b \in R$ $p r o v e t h a t \frac{π - x}{2} = a r c t a n (\frac{s i n x}{1 - c o s x})$ $2) p r o v e t h a t ∣ a r c t a n (a) - a r c t a n (b) ∣⩽∣ a - b ∣$ $3) l e t θ \in] 0, \frac{π}{2} [, x \in [θ, 2 π - θ], r \in [0, 1 [p r o v e t h a t$ $∣ φ (x, r) - \frac{π - x}{2} ∣⩽ \frac{1 - r}{{(1 - c o s θ)}^{2}}$

Question Number 35609 Answers: 0 Comments: 0

let r ∈[0,1[ and x∈ R and ϕ(x,r) = arctan( ((rsinx)/(1−r cosx))) 1) prove that (∂ϕ/∂x)(x,r) =Σ_(n=1) ^∞ r^n cos(nx) 2)prove that ϕ(x,r) = Σ_(n=1) ^∞ r^n ((sin(nx))/n)

$l e t r \in [0, 1 [a n d x \in R a n d$ $φ (x, r) = a r c t a n (\frac{r s i n x}{1 - r c o s x})$ $1) p r o v e t h a t \frac{\partial φ}{\partial x} (x, r) = \sum_{n = 1}^{\infty} r^{n} c o s (n x)$ $2) p r o v e t h a t φ (x, r) = \sum_{n = 1}^{\infty} r^{n} \frac{s i n (n x)}{n}$

Question Number 35608 Answers: 0 Comments: 0

let r∈[0,1[ and x from R F(x,r) = (1/(2π)) ∫_0 ^(2π) (((1−r^2 )f(t))/(1−2r cos(t−x) +r^2 ))dt with f ∈ C^0 (R) 2π periodic and ∣∣f∣∣=sup_(t∈R) ∣f(t)∣ prove that F(x,r)= (a_0 /2) + Σ_(n=1) ^∞ r^n (a_n cos(nx) +b_n sin(nx)) with a_n = (1/π) ∫_0 ^(2π) f(t) cos(nt)dt and b_n = (1/π) ∫_0 ^(2π) f(t)sin(nt)dt

$l e t r \in [0, 1 [a n d x f r o m R$ $F (x, r) = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - r^{2}) f (t)}{1 - 2 r c o s (t - x) + r^{2}} d t w i t h$ $f \in C^{0} (R) 2 π p e r i o d i c a n d ∣∣ f ∣∣= {s u p}_{t \in R} ∣ f (t) ∣$ $p r o v e t h a t F (x, r) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} r^{n} (a_{n} c o s (n x) + b_{n} s i n (n x))$ $w i t h a_{n} = \frac{1}{π} \int_{0}^{2 π} f (t) c o s (n t) d t a n d$ $b_{n} = \frac{1}{π} \int_{0}^{2 π} f (t) s i n (n t) d t$

Question Number 35605 Answers: 0 Comments: 0

let r∈[0,1[ and θ ∈ R,x∈ R prove that 1) 1+ 2 Σ_(n=1) ^(+∞) r^n cosθ = ((1−r^2 )/(1−2r cosθ +r^2 )) 2)1 =(1/(2π)) ∫_0 ^(2π) (((1−r^2 ))/(1−2rcos(t−x) +r^2 ))dt

$l e t r \in [0, 1 [a n d θ \in R, x \in R p r o v e t h a t$ $1) 1 + 2 \sum_{n = 1}^{+ \infty} r^{n} c o s θ = \frac{1 - r^{2}}{1 - 2 r c o s θ + r^{2}}$ $2) 1 = \frac{1}{2 π} \int_{0}^{2 π} \frac{(1 - r^{2})}{1 - 2 r c o s (t - x) + r^{2}} d t$

Question Number 35603 Answers: 0 Comments: 0

let x ∈ R and {x}=x −[x] prove that ∫_1 ^(+∞) (({x})/x^2 ) dx is convergent and find its value .

$l e t x \in R a n d {x} = x - [x]$ $p r o v e t h a t \int_{1}^{+ \infty} \frac{{x}}{x^{2}} d x i s c o n v e r g e n t a n d f i n d$ $i t s v a l u e .$

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