Show that if f(x) is integrable on every interval of real numbers, and if a and b are real numbers with a < b, then a. f(x) dr and f(x) dx both converge if and only if S f(x) dr and , f(x) dx both converge. b. fx) de + f(x) dx = f f(x) dx + f(x) dx when the integrals involved converge.

Q: (b) Prove that the improper integral Ï Ï -8 -∞ converges. dx dy (1 + x² + y2)3/2

A: Consider the given improper integral, I=∫-∞∞∫-∞∞dxdy1+x2+y232 Change into circular coordinates by…

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Q: Determine whether the integral dr converges or diverges. z(In z)3/2 If it converges, give its value.

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Q: Use the Comparison Test to determine whether or not the integral converges.

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Q: Determine whether the improper integral converges and, if so, evaluate it.

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Q: Determine whether the doubly infinite improper integral converges and, if so, evaluate it.

A: ∫-∞∞x dx1+x2

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Q: 1 (a) Use the Integral Test to show that > converges. 2. n' +1 d 1 n = 0 -1 tan dx Hint: 1+x?

A: As per our guidelines, we are supposed to solve only one question.  Kindly post other questions  as…

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Q: Let fn (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] True False

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Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

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Q: Show that f„(x) = x" (1– x") converges pointwise V x € (-1,1]. Is f„(x) converges uniformly?

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Q: x² +3 dx 2 2x* +x 00 4 1

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Q: Determine whether the doubly infinite improper integral converges and, if so, evaluate it.

A: To evaluate the improper integral, split the integral up into two separate integrals at point x=0.…

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Q: 30. Determine if the integral dx is convergent or divergent c. Neither a. Convergent b. Divergent d.…

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Q: Let S f(x) dæ = S". f(x) dx + So® f(x) dx. If S". f(x) dæ is convergent, then Select one:

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Q: (c) Show that fn (x) : x2+n? converges uniformly on [0, 0).

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Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

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Q: (b) Prove that the improper integral ]] 8888 converges. dr dy (1 + x² + y2)3/2

A: Solution: Given improper integral, ∫-∞∞∫-∞∞dxdy1+x2+y232 As we know the convergence of the series is…

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A: To solve the given improper integral.

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Q: If f(x) ≤ g(x) for all x in [a, +∞o) and converges. •+∞ f(x) da converges, then [*** g(x) dx also

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Q: determine whether the improper integral converges and, if so, evaluate it.

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Q: 2.1 Show that a sequence of Riemann-Integrable functions fn(r) converges uniformly to a…

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Q: Prove with an e – N argument that {fn(x) + gn(x)} converges uniformly to f (x) + g(x) on R.

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Q: 1 Jo 7x +9x' dx .3

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A: Given: the integral below if convergent evaluate the integral, otherwise show that it is divergent.

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Q: 1 Given f : [2, 0) → R, Prove that fn(x) = converges uniformly to f(x) = 0. 1+ x"

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Q: If fn : R → [0, 0) is a sequence of integrable functions, fn + f a.e. as n → 0, and S fn + 0. Prove…

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Q: b) Use the Comparison Theorem with f(x) = - to show whether , g(x)dx is convergent or divergent.

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Q: 2. Use the Comparison Theorem to determine whether the integral x +4 J. V-2-1 dx x4 – 2x – 7…

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A: Given, fx=nxe-nx , x∈0,∞ Since, limn→∞fx=0 , ∀x x=0, fx=0 and e-nx→0 as n→∞ Let, fx converges…

Q: determine whether the improper integral converges and, if so, evaluate it.

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Q: Consider the sequence of functions {fn}n=2 where fn: [a, b] → R defined for n > 2 by if 0 sxs fa =…

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Q: Suppose that fn(x) and f(x) are functions R –→ R. Suppose also that fn converges uniformly to f as n…

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