Question

Calculate integral from 0 to ∞ dx/ ((1 + x2) ^2) (by integrating the function 1 (1 + z 2) 2 around the upper half-circle of radius R, centered at 0, and letting R → ∞.)

Answer #1

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0 xy dy dx +Z2 √2Z√4−x2 0 xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r.
b. [4] Compute the integral.

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0
xy dy dx +Z2 √2Z√4−x2 0
xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r. b. [4] Compute the integral.

Evaluate double integral Z 2 0 Z 1 y/2 cos(x^2 )dx dy
(integral from 0 to 2)(integral from y/2 to 1) for cos(x^2) dx
dy

Integrate these
(1) Integral 1/[(ax)2- b2]3/2
dx
(2) Integral (x2 - 2x - 1) / (x -
1)2(x2 + 1) dx
I would be very thankful if the answer is back with solution
within 30mins
Thank you in advance

Evaluate the line integral by the two following methods.
(xy dx + x2 dy)
C is counterclockwise around the rectangle with
vertices (0, 0), (2, 0), (2, 1), (0, 1)
(a) directly
(b) using Green's Theorem

1. the integral of 0 to 1 of (x) / (2x+1)^3 dx
2. the integral of 2 to 4 of (x+2) / (x^2+3x-4) dx

a). Find dy/dx for the following integral.
y=Integral from 0 to cosine(x) dt/√1+ t^2 ,
0<x<pi
b). Find dy/dx for tthe following integral
y=Integral from 0 to sine^-1 (x) cosine t dt

Calculate the line integral of the vector field
?=〈?,?,?2+?2〉F=〈y,x,x2+y2〉 around the boundary curve, the curl of
the vector field, and the surface integral of the curl of the
vector field.
The surface S is the upper hemisphere
?2+?2+?2=36, ?≥0x2+y2+z2=36, z≥0
oriented with an upward‑pointing normal.
(Use symbolic notation and fractions where needed.)
∫?⋅??=∫CF⋅dr=
curl(?)=curl(F)=
∬curl(?)⋅??=∬Scurl(F)⋅dS=

(x^2+y^2)dx-2xydy=0
is nonexact
find the integrating factor

compute the definite integral. The integral of 0 to 1
x^4(1-x)^4/1+x^2 dx

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