is Theorem 1.6. The line integrals of f in D are independent of path if and only if f has an antiderivative F in D.

Q: A. Expand, that is, perform the indicated operations on the differential operator D. 1. xD(D + x) 2.…

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Q: Evaluate the line integral / (x+ 2- rom (0,0) to (3, 0), the path C2 fro co (0,0) by applying the…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

Q: The set of all values of p which makes the integral x* dx is divergent is a) (-1, 0) b) (-n,-1) c)…

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Q: 2. Evaluate each of the following line integrals in two ways∗ . (a) Z C1 F~ 1 · d~r, where F~ 1(x,…

A: Given vector field is F→(x,y)=(2x-cosy)i^+(xsiny)j^F→.dr→=(2x-cosy)dx+(xsiny)dy C1 is the…

Q: Evaluate the line integral along the given path: 1. So ry ds, where C : r(t) = 4ti + 3t²j, 0 < t <…

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Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

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Q: Use Green's Theorem to evaluate the line integral |(v - x)dx + (2x – v)dy for the given path. C: x =…

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Q: evaluate the integral. +t In(t“ + 1)j |dt te

A: Given that: ∫01t·e-t2i+tlnt2+1jdt

Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

Q: Use Green's Theorem to evaluate the integral (y - x)dx + (2x - y)dy for the given path. C: x = 2…

A: Definition used -                                According to the green theorem -…

Q: Show that the line integral | 2.xe- dx + (2y – a²e¬") dy is independent of path and evaluate the…

A: Here we have to show that the given line integral is path independent and evaluate the line integral…

Q: Evaluate the integral.

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Q: 1 et dx 1 (1 — е*)2

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Q: Evaluate |- e*dz where c is |z|=3 using Cauchy's integral formula „(1+2),

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Q: Evaluate the integral ∮C 4x3y dx + x4 dy for any closed path C.

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Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

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Q: Verify that the integral of the beta density from −qto q equals 1 for(a) α = 2 and β = 4;(b) α = 3…

A: The Beta density function is given by: We need to prove:

Q: Verify Green's Theorem by evaluating both integrals | y2 dx + x2 dy dA ду for the given path. C:…

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Q: 38. Evaluate the integral dy + x³ dx for any closed path C.

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Q: Theorem 1 : If f L[S(c); p] = F(p), then L;P = (u) đu , provided that the integral exists.

A: The solution is given as

Q: Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate…

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Q: Evaluate the given line integral. I (y, –x) · dR, where C consists of 4y = x² from (-2, 1) to %3D C…

A: Line integration of the function

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

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Q: 2. Evaluate the given line integral, (x – y, y) · dR, where C is the line segment from (0, 1) to…

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Q: Find the product of the integral 1- S x²cos(xy)dxdy

A: We have to change order of integration first.

Q: +0o 34 Evaluate the imporper integral f10 , dæ divergent 17 In 9

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Q: Evaluate the given integral where C is the directed line segment that joins the point 1 to the point…

A: The given problem is to evaluate the given path integral over given path of line segment joining…

Q: Evaluate the following line integral. x-ds; C is line segment from (3,5,6) to (- 3,12,2).

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Q: Show that the following line integral is path independent and evaluate its value. Show the details…

A: Solution:

Q: Evaluate the line integral | H. dř; Ħ = e" î + [xe" + In(z)] ĵ + 22 k, and C is the path defined by

A: Given that  H→=eyi^+xey+lnzj^+2yzk^      (1)  and r→=t,1,1, t∈0,1                (2)      from…

Q: Wyx %3D If w = e* + x In y + y In x prove that Wgy

A: Consider a multivariable function . Then the derivative of the function with respect to x is the…

Q: Evaluate the line integral H. dř; Ħ = e" ì + [æe" + In(z)] ĵ +2² k, and C is the path defined by 7…

A: Given H→=eyi^+xey+lnzj^+2yzk^ We know that dr→=dxi^+dyj^+dzk^…

Q: 7. Use the Fundamental Theorem of Line Integrals to evaluate S(6xy, 3x² + 6y²) · dr where C is any…

A: Line integral theorem

Q: Let / = [c, d] [a, b], show that g =xj is Reimann integrable on [a, b] and b. S g(x)dx = d-c a

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Q: (1) Evaluate the line integral of F (x, y) = (x – y, x²) over the line from (1,0) to (3, 2). |

A: evaluate line integral in following way

Q: Compute the path integral of F = ⟨ y , x ⟩ along the line segment starting at    ( 1 , 0 ) and…

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Q: 4. Find path integral J xdx+ ydy, if L is the segment: x-1,y= and re (0.1]

A: Given x=t, y=t2, t∈0,1

Q: 1. What should be A for the path independent of (2,4) ry +A -dr dy (1,2) Moreover evaluate the…

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Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy dA ax ду for the given path. C:…

A: This question is related to vector calculus solved in step 2.

Q: Evaluate the iterated integral. cos(s*) dt ds

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Q: (1) Show that F(x, y) : (2x + y, x - y) is path independent and use the fundamental = theorem of…

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Q: Verify Green's Theorem by evaluating both integrals [ 1² dx + x² oy = √ √ (2x - 3) da dA dy əx JR…

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Q: Evaluate the following path integrals f(x, y, z) ds, under the following cond (a) f(x, y, z) =…

A: using the formula∫c Fds =∫abF(r(t)) r'(t)dt

Q: Verify Green's Theorem by evaluating both integrals ƏN [_y² dx + x² ay = [] (ax - 3+²) da d'A dy ay…

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Q: Compute the following Cauchy principal value integral: dx р.v. x4 – 1 п/2 -T/2 Зп /2 -27

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Q: 37. Evaluate the integral fи 4x³y dx + x* dy for any closed path C.

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Q: Evaluate the integrals in ∫2 t e-t2 dt

A: Given:2∫ t e-2t dt

Q: Prove,J w' Sin wx du %3D COSZ 2

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Q: Let f(z)=4Re(z) and C be the line segment connecting 1+i to 0. Then the integral of f over C is…

A: Given function is fx=4Rez and C be the line segment connecting 1+i to 0 We have to find the integral…