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IntegrationQuestion and Answers: Page 297

Question Number 33258 Answers: 0 Comments: 0

if (1/(1+cosx)) = (a_0 /2) +Σ_(n≥1) a_n cos(nx) calculate a_0 and a_n

$i f \frac{1}{1 + c o s x} = \frac{a_{0}}{2} + \sum_{n ⩾ 1} a_{n} c o s (n x) c a l c u l a t e a_{0}$ $a n d a_{n}$

Question Number 33257 Answers: 0 Comments: 1

let g(x)= (1/(1+x^4 )) 1) find g^((n)) (x) 2) calculate g^((n)) (0) 3) if g(x)=Σ u_n x^n find the sequence u_n

$l e t g (x) = \frac{1}{1 + x^{4}}$ $1) f i n d g^{(n)} (x)$ $2) c a l c u l a t e g^{(n)} (0)$ $3) i f g (x) = Σ u_{n} x^{n} f i n d t h e s e q u e n c e u_{n}$

Question Number 33222 Answers: 0 Comments: 0

let give n ≥3 integr calculate I_n = ∫_(−∞) ^(+∞) (dx/(1+x +x^2 +....+x^(n−1) ))

$l e t g i v e n ⩾ 3 i n t e g r c a l c u l a t e$ $I_{n} = \int_{- \infty}^{+ \infty} \frac{d x}{1 + x + x^{2} + . . . . + x^{n - 1}}$

Question Number 33223 Answers: 0 Comments: 0

let A_n =∫_(−∞) ^(+∞) (e^(iπx) /(1+x+x^2 +...x^(n−1) )) with n≥3 integr find the value of A_n .

$l e t A_{n} = \int_{- \infty}^{+ \infty} \frac{e^{i π x}}{1 + x + x^{2} + . . . x^{n - 1}} w i t h n ⩾ 3 i n t e g r$ $f i n d t h e v a l u e o f A_{n} .$

Question Number 33210 Answers: 0 Comments: 0

find lim_(x→+∞) x e^(−x^2 ) ∫^(x−1) _0 e^t^2 dt

$f i n d {l i m}_{x \to + \infty} x e^{- x^{2}} {\int_{0}}^{x - 1} e^{t^{2}} d t$

Question Number 33204 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cos(ax))/(1+x+x^2 )) dx.

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

Question Number 33232 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((x sin(2x))/((1+4x^2 )^2 )) dx .

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

Question Number 33202 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) (dt/((1+t +t^2 )^2 )) .

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

Question Number 33175 Answers: 0 Comments: 1

find ∫_0 ^1 (dt/((1+t^2 )^2 ))

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

Question Number 33172 Answers: 0 Comments: 0

find ∫ (dx/(x +(√(x^2 −3x+2)))) .

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

Question Number 33170 Answers: 0 Comments: 1

prove that ∫_0 ^∞ ((∣sinx∣)/x) dx is divergent.

$p r o v e t h a t \int_{0}^{\infty} \frac{∣ s i n x ∣}{x} d x i s d i v e r g e n t .$

Question Number 33169 Answers: 1 Comments: 1

find the value of ∫_0 ^π (dx/(1+2 sin^2 x)) .

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

Question Number 33166 Answers: 0 Comments: 0

find the value of ∫_0 ^1 (dx/(1+x^4 )) .

$f i n d t h e v a l u e o f \int_{0}^{1} \frac{d x}{1 + x^{4}} .$

Question Number 33155 Answers: 0 Comments: 4

Evaluate ∫_(−∞) ^∞ 3x^2 (x^3 + 1)^2 e^(−x^6 − 2x^3 ) dx

$Evaluate$ $\int_{- \infty}^{\infty} 3 x^{2} {(x^{3} + 1)}^{2} e^{- x^{6} - 2 x^{3}} d x$

Question Number 33130 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((1+x cosθ)/(x^2 +2x cosθ +1)) dx .

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

Question Number 33129 Answers: 0 Comments: 2

1)find the value of u_n =∫_(−∞) ^(+∞) ((cos(nx))/(4 +x^2 )) dx 2) find the nature of Σ u_n .

$1) f i n d t h e v a l u e o f u_{n} = \int_{- \infty}^{+ \infty} \frac{c o s (n x)}{4 + x^{2}} d x$ $2) f i n d t h e n a t u r e o f Σ u_{n} .$

Question Number 33128 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )( 1+x^4 ))) .

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

Question Number 33120 Answers: 1 Comments: 0

let give α>0 find the value of ∫_0 ^1 (dx/(√((1−x)(1+αx)))) .

$l e t g i v e α > 0 f i n d t h e v a l u e o f \int_{0}^{1} \frac{d x}{\sqrt{(1 - x) (1 + α x)}} .$

Question Number 33119 Answers: 0 Comments: 1

find ∫_0 ^∞ (t^n /(e^t −1)) dt by using ξ(x) for n integr ξ(x)=Σ_(n=1) ^∞ (1/n^x ) with x>1 .

$f i n d \int_{0}^{\infty} \frac{t^{n}}{e^{t} - 1} d t b y u s i n g ξ (x) f o r n i n t e g r$ $ξ (x) = \sum_{n = 1}^{\infty} \frac{1}{n^{x}} w i t h x > 1 .$

Question Number 33069 Answers: 0 Comments: 0

by using residus theorem prove that ∫_0 ^∞ (t^(a−1) /(1+t)) dt = (π/(sin(πa))) with 0<a<1 .

$b y u s i n g r e s i d u s t h e o r e m p r o v e t h a t$ $\int_{0}^{\infty} \frac{t^{a - 1}}{1 + t} d t = \frac{π}{s i n (π a)} w i t h 0 < a < 1 .$

Question Number 33028 Answers: 0 Comments: 0

find the value of∫_0 ^∞ (e^(−[t]) /(t+1))dt .

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{e^{- [t]}}{t + 1} d t .$

Question Number 33027 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (x^3 /(1+x^5 ))dx.

$c a l c u l a t e \int_{0}^{\infty} \frac{x^{3}}{1 + x^{5}} d x .$

Question Number 33026 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((1+x^4 )/(1+x^6 )) dx .

$c a l c u l a t e \int_{0}^{\infty} \frac{1 + x^{4}}{1 + x^{6}} d x .$

Question Number 33009 Answers: 2 Comments: 1

help ! ! ! ∫ (dx/(csc(x)−1)) = ? [ my way ] ∫( (dx/((1/(sinx)) − 1)) ) =∫((sinx)/(1−sinx)) dx =−∫ ((sinx−1+1)/(sinx−1)) dx =−∫1+(1/(sinx−1)) dx =−(∫1dx+∫((sinx+1)/((sinx−1)(sinx+1))) dx) =−(x+C−∫((sinx+1)/(1−sin^2 x)) dx) =−(x+C−∫ ((sinx)/(cos^2 x)) dx−∫ (1/(cos^2 x)) dx) =−(x+C+∫(cosx)^(−2) dcosx−∫(1/(cos^2 x))dx) =−(x−(cosx)^(−1) +C−∫(1/(cos^2 x))dx) ...and I can′t solve the ∫(1/(cos^2 x))dx oh i just found that is tanx+C

$h e l p!!!$ $\int \frac{d x}{c s c (x) - 1} = ?$ $[m y w a y]$ $\int (\frac{d x}{\frac{1}{s i n x} - 1})$ $= \int \frac{s i n x}{1 - s i n x} d x$ $= - \int \frac{s i n x - 1 + 1}{s i n x - 1} d x$ $= - \int 1 + \frac{1}{s i n x - 1} d x$ $= - (\int 1 d x + \int \frac{s i n x + 1}{(s i n x - 1) (s i n x + 1)} d x)$ $= - (x + C - \int \frac{s i n x + 1}{1 - {s i n}^{2} x} d x)$ $= - (x + C - \int \frac{s i n x}{{c o s}^{2} x} d x - \int \frac{1}{{c o s}^{2} x} d x)$ $= - (x + C + \int {(c o s x)}^{- 2} d c o s x - \int \frac{1}{{c o s}^{2} x} d x)$ $= - (x - {(c o s x)}^{- 1} + C - \int \frac{1}{{c o s}^{2} x} d x)$ $. . . a n d I {c a n}^{'} t s o l v e t h e \int \frac{1}{{c o s}^{2} x} d x$ $o h i j u s t f o u n d t h a t i s t a n x + C$

Question Number 32994 Answers: 0 Comments: 1

find ∫_0 ^∞ ((1−cos(λx))/x^2 ) dx with λ>0 .

$f i n d \int_{0}^{\infty} \frac{1 - c o s (λ x)}{x^{2}} d x w i t h λ > 0 .$

Question Number 32993 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−λx) ((sinx)/(√x)) dx wih λ>0 .

$c a l c u l a t e \int_{0}^{\infty} e^{- λ x} \frac{s i n x}{\sqrt{x}} d x w i h λ > 0 .$

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