29. For |x|<1, prove that 16x 24x 8x 1 + 1-x1+x 1-x ..... .. 8x 8x = 1+ (1- x)? (1+x)²' (1-x)?

Q: Evaluate the improper integral 2x dx, if it exists. (x² + 81)2

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Q: OLet @ = (0,1)n Q = {x€ @104 x<1}. Prove that %3D 2) 2000m

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Q: Juneten and be fure at lebel aut least thre f (n) : Su

A: As asked we can find the answer as below

Q: Sin (n+))x Conv. unif nナノ on Prablem (5) Is the senies

A: The given series is: ∑n=1∞sinn+1xn+1 The given interval is: 0,π

Q: 7. Find the area between f(x)=-x² + 4x and g(x) = x² - 6x +5 over the interval 0≤x≤1 Ei..

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Q: Compute for the value of tanh (1 – i) Iny + 2.13i In 2 + 2.13i 2 In 5 - 4.25i 2 In v2 + 2.13i

A: To Compute: tanh-11-i Concept: 1. tanh-1z=12ln1+z-ln1-z2. lnz=lnr+iθ+2nπwhere,…

Q: 2. Use the Definition of Residue to show that: dz -ni 3 Jizl=1z² sinh z Where 0 < |z| < T.

A: Image is attached with detailed solution.

Q: Hiw:- Find Sanig ex Pansiom for:- Fouiar -\くxくo 0くXく」

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Q: 4) Evaluate the definite integral: S, (6x² – 3x + 7)dx.

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Q: 1. If f(x) dx = 3, evaluate [[1 + 21(x)] dx using the properties of interga 2. If f(x) dx = -2 and…

A: As asked we can answer question no 1 and 4 as below.

Q: find the critical points:

A: Given function is  and the interval isThe critical points of a function are obtained when the first…

Q: Evaluate the definite integral. Sí f (x) dx, ƒ (x) =3"–(15x³+ 6)/x²

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Q: 4. 13x²-1 3 X-X -dx

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Q: 13)Evaluate 13 Evaluate the integral s 5x--Xt27 dx x (x²+G) -1 3 tan 3+C b)(-ト2フx)、1mlxCx?ss)1c Sx3…

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Q: Prove that for all r € [1, 00), – sinh (- |cosh-(x)|) = vVI² – 1.

A: We know that: Cosh-1x=logex+x2-1  and  Sinhx=ex-e-x2. And, Sinh-x=-Sinhx. Therefore,…

Q: Evaluate the improper integral 2x dx, if it exists. (x² + 81)² 8.

A: The value of the integral ∫-∞02xdxx2+812 is given by: ∫-∞02xdxx2+812=-1x2+81-∞0=-181-1-∞=-181-0=-181

Q: a) For 0 ≤ x ≤ 1, show that < x². √2 + x b) Use the estimate in a) and integrate it to prove that…

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Q: 21) Find the number of compositions M needed so that the error of the integral 2.2 In(x + 2) dx is…

A: Simpson's rule can be used to approximate the value of the integration using numerical integration.…

Q: Show dz 4 In= 는I미페 - for a<0. x <

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Q: Evaluate the integral. 7 √49-x² √49-x²-22 49-x² 1 √²-3³ (x² + y² + 2²)1/2 dy dz dx =

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Q: 1 Evaluate the improper intergal | dx (x2 + 9)° using the residue theorem.

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Q: 1. (a) Prove the following. | (In x)"dx = x(In x)" – n (In x)-' dx

A: To prove that 1.a. ∫lnxndx=xlnxn-n∫lnxn-1dxb. Using 1.a compute the following      ∫lnx3dx

Q: 4. Using the definition of sech x ,x 0 in terms of e , prove that 1+V1-x2 sech x = In ,0<x <1 Hence,…

A: 4) i)

Q: Find the root of f(x) = (x – 4)° (x +2)= 0, using the initial guesses of X2 = -2.5 and Xy = -1.0,…

A: Note:- For this problem any particular method is not mentioned. so, here i'm using Bisection Method.…

Q: To prove that 2 dx 2 In V1 + (In x)² + In x| + C X/1+ (In x)² we use the substitutions

A: The given integral is ∫2 dxx1+lnx2 Find your answer below

Q: 16. Find he exact soution on The interval [0,2m] 25in?x ャ Bsinx +3= 0 5sinx t

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Q: 2) evaluate the indefinite integral £x² 9+X² dx

A: The main objective is to evaluate the integral. The integral formula is ∫xndx=xn+1n+1+C, where C is…

Q: 7. Use integration by parts to prove that for any positive integer n, п - 1 Scos | cos"x dx = 1…

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Q: 5 Only one of these calculations is correct to evaluate the definite integral 7x In(x) dx Using the…

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Q: compute the definite integral.

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Q: Show that integral 0 to infinity  x2 e -x2  dx = 1/2 integral 0 to infinity e -x2  dx

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Q: Prove that x" dx „n+1 + C, n + 1 for n + -1.

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Q: 1. Evaluate the definite integral S ,2 C (9x² + 5)dx by the limit definition.

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Q: Evaluate the integral (x – 1)ez xp- x² – 2x + 2

A: The objective is to find ∫-∞∞(x-1)eix(x2-2x+2)dx It is known that, eix is an increasing function.…

Q: 4. Show that In sin 6 d0 = – n 2 by applying the mean-value principle to In |1+ z| for |2| <r < 1,…

A: ∫0π2logsin(θ)dθ=12∫0πlogsinθdθ                         =12∫0π2logsin2θdθ…

Q: - Prove using integrals Proper tos : e even sin()fc) dx=0 -b 9- For even functions.

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Q: In(x) -dx Evaluate the integral m(2)

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Q: 2 Evaluate the improper integral 4x e dx, if exists. 8.

A: Given that,           ⇒           ∫-∞2e4x dx   ,   if exist             ⇒         find the…

Q: 4. Prove 1 < | V1+ a dx <2 0,

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Q: 19. f(x. y) In(1 - x - y) %3D

A: As per our company guidelines we are supposed to answer only the first question that you have…

Q: – 30 - Evaluate the integral: -d.x x2 Vx2 + 16

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Q: (4.4) For #27-28, find the value of c guaranteed by the Mean Value Theorem for Integrals. 27. f (x)=…

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Q: 6 Evaluate the improper integral, if exists : dx (5 – x)3 -00

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Q: In x + In(x – 1) = 10 In(x + 25) – In(x – 1) = 3 e2x – 2ex = 15 e2x – ex – 12 = Q %3D %3D -

A: i) ln x+ln(x-1)=10ii) ln(x+25)-ln(x-1)=3iii) e2x-2ex=15 Note:      Multiple part question are given,…

Q: 10 Evaluate the integral f 2x (x2 + 4) " dx using the substitution u = x² + 4. 2x (x² + 4)*" dx =|…

A: The given integral ∫2xx2+410dx

Q: 3 Find absolute extrema for fCx) = x°-9x´+15x -3 %3D in terval [o,3] Oへ

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Q: 1. Evaluate the integrals that converge: dx 2x ! dx (x2 + 5)3 b) a) 3-x - 00

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Q: Q2-Prove that = va* – x² ify= ()sin () + () va² - x* = va² – x* if y = (;")sin*+ (4) + () Va² - x²

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