d x"J,(x)=-x"Jns(x) d, n+1 X.

Q: Define f(x) = x if x is rational and f(x) = 0 if x is irrational. Compute So f dx and ſ f dx. Is f…

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Q: 2. Let f and g be integrable functions such that f(x) dx = 6 and g(x) dx = –2. Using the Linearity…

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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: Discuss limit(n →∞)fn on B and C for  fn(x)=x/n; B=E1; C = [a, b] ⊂ E1 fn(x)=(cos x + nx)/n; B=E1

A: Note: Since you have not mention anything about the set E1, We are considering it any sunset of real…

Q: 10 Suppose that " fx) dx and Ax) dx = -5, and "g(x) dx = -1 and g(x) dx = 3. Compute the given…

A: Given,∫04f(x)dx=7∫02f(x)dx=-5∫04g(x)dx=-1∫02g(x)dx=3

Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

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Q: f is not integrable on [0, 1].

A: It is given that f:[0, 1]→ℝ by f(x)=5x  if x∈ℚ0    if x∉ℚ We know, if lower integral and upper…

Q: Recall that 1, x >0 d (]xl) dx —1, х<0 x < d Use this result and the chain rule to find (sinx|) for…

A: Here, recall that  -π,-π2 is 3rd quadrant, where sin is -ve-π2,0 is 4th quadrant, where sin is…

Q: If f : [0,2] → R is defined by f(x) = 1 if 0 ≤ x ≤ 1, f(x) = 2 if 1 < x ≤ 2. Show that f is…

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

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Q: Find the Fourier integral of f(x) = e-k* when x>0 and f(-x) = f(x), hence prove that cos(wx) dw w…

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Q: If f : [0,2] → R is defined by f(x) = 1 if 0 ≤ x ≤ 1, f(x) = 2 if 1 < x ≤ 2. Show that f is…

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Q: 2. Find an asymptotic expansion of the solution y(x) to leading order that is valid on [0, 1] for…

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Q: Consider the definite integral. (8x- – 1) e*x²-x dx Let u = 4x2 - x. Use the substitution method to…

A: Answer and explanation is given below

Q: 2. a) Compute the integral fa f(x) dx using 5 point composite rectangle, trapezoid and Simpson rules…

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Q: 1. Use the Fundamental Theorem of Calculus to evaluate the following (DO NO 5x -6x dx 1 | cos (x)--…

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Q: f (x) = = cos 2x; [0,T] G and

A: topic - mean value theorem

Q: Evaluate the integral. 5n /2 2, 1 - cos t dt 3π/2

A: ∫3π25π21-cos2tdt=∫3π25π2sin2tdt=∫3π25π2sintdt=∫3π22π-sintdt+∫2π5π2sintdt=1+1=2

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

A: Prove,∫z=1dzz2 sinh z=-πi3 ∫z=1dzz2 sinh z=∫z=1fz dz .........equation 1 fz=1z2 sinh z , 0&lt;z&lt;π…

Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

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Q: 68. Show that the asymptotic representation of the Bessel function Jn (kr) for large kr is Jn (kr) =…

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Q: Consider the function f(x) = lIn(x). Show that x(In(x) - 1) is an antiderivative of the function f.…

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Q: f(x) = In(2 + cos x). -2.125, xk+1 = Ik +0.5, yk = f (xk), k E N To = %3D %3D = 0r

A: The function is fx=ln2+cosx Conditions: x0=-2.125, xk+1=xk+0.5, yk=fxk, k∈ℕ xk+1=xk+0.5…

Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

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Q: | 45. Apply integration to the expansion 00 =E(-1)"x" = 1- x+x – x +.. 1+x n=0 to prove that for -1…

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Q: 2n Jnsi (x) = " J,(x) – J(4)

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Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

Q: 1. Evaluate the indefinite integral. Hint: U-substitution. 3 dx x In x

A: ∫ 3x lnx dx 3∫ 1lnx·1x dx

Q: 2. Evaluate the following integral using the Fundamental Theorem of Calculus. (t-3 – 8) dt

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Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A: ddxsinnx =cosnxddxnx =ncosnx

Q: Q1) suppose that W= x2+y2+z² and x cos 0, y = sin0, z = tan 0 ,use the chain rule to find de when 0

A: We have, W=x2+y2+z2 and x=cosθ, y=sinθ, z=tanθ  We have to find, dwdθ=? at θ=π4

Q: Let G(x) = tan 9t dt. In order to use the Fundamental Theorem of Calculus (Part 2), G(x) needs to be…

A: Let a,b&gt;0 with b&gt;a Then we can write, ∫abf(x) dx as ∫0bf(x) dx - ∫0af(x)dx

Q: Transform Bessel's equation x²²y + xdx+(x²-n²) y=0 by substitution x== souber moitops navig Hence…

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Q: |1) Use a sine expansion to compute the solution of ( ди д'и at ôx? u(0,t)=u(1,t) = 0,t > 0, u(x,0)…

A: According to the given information, it is required to compute the solution of the given equations.

Q: The modified Bessel function I,6x) has the integral representation /2 2 1(x) = cosh(x sin 0 )d0. Use…

A: We will use given value of x and mentioned process ( trapizoidal / simpson ) to find value of…

Q: 1) If f(x)=x2, find f(x+2)-f(2) .what is bijective h mapping? 2) Does the integral J, dx exist?

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Q: A:= {(x, y) € R² :1< log(x) < 2}. = {(x, y, z) E R³: sin(xy² + «/2) · cos(z) = 1} . (a) Decide…

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Q: 11. Show that the definite integral of f(x) = c over [a, b] is c(b-a) by applying the definition of…

A: Here given function as fx=c where the function fx is defined on the interval a, b.

Q: Prove that the Generating function of Bessel function i.e. £1J„(x)t" = e*/2(-1/2), %3D

A: Let ϕx,t=ex2t-1t. Let us prove that ϕx,t is the generating function of the Bessel function…

Q: Let f:R^2 --> R^3 be given by f(x1,x2)=(e^(x1+x2^2, x2+cos(x1,x2), x1^2+x2^2+2) Compute the…

A: Using jacobian matrix,

Q: Evaluate the improper intergal 1 dx using the residue (x² + 1)3 theorem.

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Q: Consider the definite integral. (18x – 1) e9x²-x dx Let u = 9x? - x. Use the substitution method to…

A: We have to find the integrand after substitution.

Q: Show that the functions In x , In(x2) linearly dependent (0, co) are on

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Q: Use the second fundamental theorem of calculus along with the chain rule to find d sint dt dx 1 o…

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Q: Evaluate Poisson Integral: In(1-2a cos(x) + a²) dx ,a.) [a| 1,

A: To evaluate Poisson's integral: ∫0πln1-2αcos(x)+α2dx  a)α&lt;1 b)α&gt;1

Q: Suppose that f is continuous on [0, 1], ƒ(0) = 0, and f'(0) exists. Prove that the Lebesgue integral…

A: This is a question of measure theory.

Q: 4: Find the indefinite integral of f(u) = -1-3√ =

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.