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IntegrationQuestion and Answers: Page 298

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 .

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

Question Number 33125 Answers: 0 Comments: 0

let give u_n = ∫_0 ^π ((cos(nx)dx)/(1−2λcosx +λ^2 )) 1) prove that λ u_(n+2) −(1+λ^2 )u_(n+1) +λ u_n =0 2) ptove that Σ u_n is convergent and find its sum

$l e t g i v e u_{n} = \int_{0}^{π} \frac{c o s (n x) d x}{1 - 2 λ c o s x + λ^{2}}$ $1) p r o v e t h a t λ u_{n + 2} - (1 + λ^{2}) u_{n + 1} + λ u_{n} = 0$ $2) p t o v e t h a t Σ u_{n} i s c o n v e r g e n t a n d f i n d i t s s u m$

Question Number 32958 Answers: 0 Comments: 1

Question Number 32951 Answers: 2 Comments: 1

Evaluate ∫((x^4 +1)/(x^6 +1))dx [W.B.H.S 2018]

$E v a l u a t e$ $\int \frac{x^{4} + 1}{x^{6} + 1} d x [W . B . H . S 2018]$

Question Number 32939 Answers: 1 Comments: 1

1) study the convergence of ∫_0 ^1 (x^p /(1+x)) dx 2) find lim_(p→∞) ∫_0 ^1 (x^p /(1+x))dx .

$1) s t u d y t h e c o n v e r g e n c e o f \int_{0}^{1} \frac{x^{p}}{1 + x} d x$ $2) f i n d {l i m}_{p \to \infty} \int_{0}^{1} \frac{x^{p}}{1 + x} d x .$

Question Number 32928 Answers: 2 Comments: 2

plz help Evalute ∫_(π/3 ) ^(π/4) ((sin^2 x)/(√(1−cosx)))dx

$p l z h e l p$ $E v a l u t e$ $\underset{π / 3}{\int^{π / 4}} \frac{\sin^{2} x}{\sqrt{1 - c o s x}} d x$

Question Number 32789 Answers: 0 Comments: 0

∣∫_a ^b f(x)dx≤∣∫_a ^b ∣f(x)∣dx∣

$∣ \underset{a}{\int^{b}} f (x) d x ⩽∣ \underset{a}{\int^{b}} ∣ f (x) ∣ d x ∣$

Question Number 32785 Answers: 1 Comments: 0

∫((3x^2 +2x−4)/(7x^2 −9x+2))dx

$\int \frac{3 x^{2} + 2 x - 4}{7 x^{2} - 9 x + 2} d x$

Question Number 32763 Answers: 1 Comments: 1

Question Number 32741 Answers: 0 Comments: 0

find ∫_0 ^1 ((ln(t^2 +2t cosx +1))/t)dt .

$f i n d \int_{0}^{1} \frac{l n (t^{2} + 2 t c o s x + 1)}{t} d t .$

Question Number 32740 Answers: 0 Comments: 2

find∫_0 ^∞ ((ln(x^2 +t^2 ))/(1+t^2 ))dt

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

Question Number 32739 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ (e^(−t) /(1+xt))dt calculate f^((n)) (0).

$l e t f (x) = \int_{0}^{\infty} \frac{e^{- t}}{1 + x t} d t$ $c a l c u l a t e f^{(n)} (0) .$

Question Number 32737 Answers: 1 Comments: 0

let give 0≤x≤1 calculate ∫_0 ^∞ ((arctan((x/t)))/(1+t^2 )) dt

$l e t g i v e 0 ⩽ x ⩽ 1 c a l c u l a t e \int_{0}^{\infty} \frac{a r c t a n (\frac{x}{t})}{1 + t^{2}} d t$

Question Number 32736 Answers: 0 Comments: 0

let o≤x≤1 find ∫_0 ^x ((lnt)/(t^2 −1))dt

$l e t o ⩽ x ⩽ 1 f i n d \int_{0}^{x} \frac{l n t}{t^{2} - 1} d t$

Question Number 32733 Answers: 0 Comments: 0

prove that Σ_(n=0) ^∞ (1/((n!)^2 )) =(1/(2π)) ∫_0 ^(2π) e^(2cosx) dx .

$p r o v e t h a t \sum_{n = 0}^{\infty} \frac{1}{{(n!)}^{2}} = \frac{1}{2 π} \int_{0}^{2 π} e^{2 c o s x} d x .$

Question Number 32731 Answers: 0 Comments: 0

1) prove that ∫_0 ^1 ((arctant)/t)dt=−∫_0 ^1 ((lnt)/(1+t^2 ))dt 2) find ∫_0 ^1 ((arctant)/t)dt at form of serie

$1) p r o v e t h a t \int_{0}^{1} \frac{a r c t a n t}{t} d t = - \int_{0}^{1} \frac{l n t}{1 + t^{2}} d t$ $2) f i n d \int_{0}^{1} \frac{a r c t a n t}{t} d t a t f o r m o f s e r i e$

Question Number 32729 Answers: 0 Comments: 0

find lim_(n→∞) ∫_0 ^n (cos((x/n)))^n^2 dx.

$f i n d {l i m}_{n \to \infty} \int_{0}^{n} {(c o s (\frac{x}{n}))}^{n^{2}} d x .$

Question Number 32724 Answers: 0 Comments: 0

let A_n = ∫_0 ^n (√(1+(1−(x/n))^n )) dt. find a rquivalent of A_n .

$l e t A_{n} = \int_{0}^{n} \sqrt{1 + {(1 - \frac{x}{n})}^{n}} d t .$ $f i n d a r q u i v a l e n t o f A_{n} .$

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