7. a) Prove that for r > 0 * (E) + c. Vr2 – r²dx = vr² – x² + -1 sin 2 +C.

Q: Locate the positive root of f(x)= sin(x) + cos(1+x²)-1 using bisection method with initial guesses…

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Q: 2.3 Find the general solution of p fanx+q tany =lanz

A: Given:   P tanx+q tan y=tanz→           1   is of the form pP+qQ=z:

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: Evaluate the integral 5a2 - x+27 da. (x2 +9) 5x3 x2 + 27z) · In|r(x² + 9)|+ C 2 3 In |피 + 2z In |2…

A: Topic - indefinite integration  Note- as per our guideline if more than one question uploaded only…

Q: 4. (a) (b) Approximate the integral f x² cos xdx with n = 6 using the composite trapezoid rule;…

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Q: 7 y=tanx show thats y" = -2Siny cosy 1f y=e Snbx show that:

A: Dear Student, according to the Bartleby guidelines, we can answer only first question if multiple…

Q: 2. If z = x° In 2.xy, find at 1,-

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Q: Q6:-Prove that: y" = 1 if y=;csc (tan"

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Q: iii) E(-1)"+1 cos²(÷) COS n=1

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Q: 4 – dt 2 cos 4t 1.

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Q: 4. If x = cose + isin0 & 1 + V1– a² = na then prove that 1+ acose =(1+ nx)(1+) %3D 2n

A: Given, x= cos θ+i sinθ and na=1+1-a2      To prove:  1+a cosθ=a2n(1+nx)(1+nx)       Let  x=…

Q: S Prove that L(cos 3t) : %3D (s2+9)

A: We know that Lf(t) = ∫0∞e-stf(t) dt Then Lcos 3t = ∫0∞e-st cos 3t dt                             (1)…

Q: Compute f'cx) of ×そ9 O f(x)- X-3 at X, = 3 の f(x)= x*+ ラメ+ タメ+1 at Xo=-2 f(x)= 4cos x + 2 sinx at X=…

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Q: 2- Find the largest possible interval in which the given DE has a unique solution tan(x)y" + xy'-y =…

A: Given equation is, tanxy''+xy'-y=x3 ,x0=-2,x0=5 We can write the given equation as,…

Q: S Prove that L(cos 3t) (s²+9)

A: Given L(cos3t)

Q: In(1+2x) Find the Padé approximation R1,2(x) for f(r) %3D

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Q: B- Find the solution to the equation = k satisfying the conditions at ax2 1-- 30 at x 0 for all…

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Q: 28- For What Values of a anel k is a X <-1 ー<くく -2 Cx) = ax-b 3 Continuons at every X ?

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Q: Which of the following is true? ® Ssin(3x)cos(4x)ax =S sih(7x)= sin(x) A 2 ® Jsin(4x)cos(3x)ax = […

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Q: 4. Find the Cauchy principal values of the following integral: dx. (x²+1)(x²+2x+2)

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Q: The limit in lim (46 x; - 28 sin (x;)) Ax i=1 on the interval o, is equal to 2 (23 x² + 28 cos (x))…

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Q: 5. Show that cos x = x has a solution in the interval [0, 1]. Hint chow that f (x) = x cos x has a…

A: Given, Cosx =x

Q: Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the…

A: We can prove given result by mathematical induction: Define a statement p(n).  Show that p(n) is…

Q: (22 – 2) | Evaluate dx 2+2 (x4+5x2 +4) tan1 2 A - BI E E E E X O Type here to search 马

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Q: Evaluate the integral 3 2x cos(xy)dydx 2

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Q: (b) f(z) = tan¯"(5z) - (In(2"))®

A: We have to find derivative

Q: 3. Use the IVT to prove that y = x ln x has at least one zero on 2'2

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Q: If fufe.- fuave diffa Veubable at'xo eI fuave difle Veubable at Xo eI Pi:I R,ists-n prove O…

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Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: In x – In c – 2 1 +- sin 2 - = 0 - 4x In-- 2y +x sin- 2y = 0 X -

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Q: 28. Evaluate f 5 sin 3x dx 2+cos 3x a. 2 In 15 sin 3x| + C b.- In 12+cos 3x|+C c. In 15 sin 3x| + C…

A: Evaluate integral

Q: (B)JE w = x*+y+z° and x = PSin ø cos@ »y=pSing Sino and z- Pcoseo prove that -2P Cos 2Ø and' %3D

A: Given  w=x2+y2-z2x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ

Q: 3.12 Еxpress lim n00 n k3D cos 1+ as an integral. (Be sure to include the lower and upper limits|of…

A: Given: limn→∞ 2n∑k=1ncos1+2kn To convert this into the integral , use the following method

Q: 1,1Y= 4 in (In (in(sec(x)) then Ý -14 (tan)! In ((In (sec)) In (sex);| O True O False

A: This is the question of the application of differential calculus.

Q: 8. 6x° cos (x³) dx

A: ∫ x8cosx3dx

Q: 2.Evaluate the following integral: √3x² cosxdx (d) single application of Simpson's 3/8 rule.

A: Here we can take N=6 ,that is number of interval =6. Then we can use Simpson's 3/8 rule to find…

Q: 6) Find most general antiderivative of the fxn ) f(2) = secx (secx +tane) Secx +tanx, = secx COS X

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Q: 7. | (x² + + 2x) cos x dx

A: Apply the integration by parts in the given integral. Let  u=(x2+2x) ; dv=cosxdxThen,du=2x+2dx ;…

Q: 1 Evaluate | (x²sin y- xcos(y))dy O 2sin y-3cosy + C 6 -3sin y+2cosy +C 6 x2(1- cos(1)) – xsin(1) +…

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Q: d 1 Prove that: dz (tan¬1 z) = 1+z?

A: Given query is to find the derivative of the expression.

Q: If u(x, y) = 2 tan-1 2 and v= In (x² + y?) , prove that: a2v = 0 дхду dx2

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

A: Cauchy residue theorem is one of the method to evaluate the integral of analytic functions.…

Q: Q6:- Prove that: y" = 0 if (cos y3csc COS

A: To prove : y'' =0

Q: 19. Consider the integral f√x²+25 dx. 9x² a. In/√x² + 25 +x/- - b. x+ In/√x² + 25/+ √x²+25 9x √x²+25…

A: Solve the integral  √(x2+25)/9x2

Q: 5) why is x3- 3x +I X +2 Conti nuous (0,00) on where is it discontinuous?

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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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Q: Qし If xと -l Sx S2 fa)= { 5-x fo) = 2< x S3 - Is fox) Condinuons at x =2 2 and sketch this funclion

A: Hello, Thank you for the question. According to our honor code can answer only one question at a…

Q: (d) Show that there is some x with 0 < x < 2 such that x2 + cos(ax) = 4. %3D

A: Solution-