Q.1 Evaluate the given integrals using part I of the Fundamental Theorem of Calculus. [3 Marks] i. (2xvx )dx dx V3 ii. -VZ xVx2-1

Q: Find the indefinite integral.

A: Given: I=∫5t+63dt  ……1

Q: Q1 - Evaluate the following integral: J (2x2-3x+2) dx (x-1)2 (x-2)

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Q: Use the fundamental theorem of calculus part 2 to evaluate the 3 integral: S (9x – 4) dæ. -2

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Q: F(4) = [, * G&+2) dt (3t +2) dt

A: Consider  F(x)=∫2x+2(3t+2)dt

Q: 3.Evaluate the following integrals by any method of your choice S(1+) ) dt a. 6x2 b. (4х3 - 9)3

A: We have to find integral

Q: 5. Evaluate the definite integral (1+Inx) а. b. S dx. 2x+3

A: a) ∫1e1+ln x2xdxb) ∫-1112x+3dx

Q: G.. Evaluate the fallowing real integrals by residue. theerem 3 -bm e 응) co3 mx (x4a?) Cx+b?)

A: This is complex analysis problem, we apply the residue theorem to solve the problem.

Q: In(x) Differentiate the function F(x) = | cos(e') dt %3| 1/r Show and justify your work. (Hint:…

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Q: Evaluate the integrals ∫√t sin (2t3/2) dt

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Q: Evaluate the indefinite integral. 4 dt (t + 11)? + C

A: Integration of 4/(t+11)^2.

Q: T 3. fz sin4 y cos7 y dy 2

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Q: Use the Fundamental Theorem of Calculus to evaluate the following definite integral. V3 2 8 dx V1-x?

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Q: Q.1) Evaluate the following integral 2dx x (3(ln z)2 + 12 In r- 15)

A: The given integral is: I=∫2 dxx3 (ln x)2+12 ln x-512

Q: 1. ſ (x-1)dx x2+5x+6

A: The given integration is ∫(x-1)x2+5x+6 dx  We have to find the integration for the given integral

Q: 2. Evaluate the Integrals using the Fundamental Theorem of Calculus. 1. S/4 cos x dx π/4 2. ¹₁ sec²…

A: First I have written the Fundamental theorem of Calculus, using it for given integral and find its…

Q: Find the indefinite integral. 3 8 dt = t3 t2 +C

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Q: Evaluate the definite integral (from 5 to 7)∫(d/dt (2+2t^4)^(1/2) dt using the Fundamental Theorem…

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Q: Find the integral. J 3 x√9 - (In x)² + C dx

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Q: 6. Evaluate the fallowing real integrals by residue theerem ? dX ニ ans : こ。(X+a?)2 493 • I- ans B 12…

A: Let's find.

Q: 50. f(x) =COS 2, Cos z, 3'3 1/3

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Q: Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the…

A: y=x24, 0, 6 The Mean Value Theorem for Integrals is, If f is continuous on a, b then there exist a…

Q: b) Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. 2 i) S 4x(1 – x³)dx…

A: (i). ∫-124x1-x2dx The antiderivative of 4x1-x2 or 4x-4x3 is 2x2-x4 Therefore,…

Q: 5. Evaluate the integrals: a) f(x²-y²)dA where D={(x,y) | -1≤x≤1, -2≤y<2}

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Q: b) Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. 2. L, 4x(1 – x²)dx…

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Q: Consider the following propositions associated with the study of the +∞ dx J - 1.1² √6 +1 improper…

A: Given : ∫1+∞dxx  in which 1x6+1≥1x

Q: Q1: Verify that the given functions are solutions of the corresponding integral equations by using…

A: Given: ₵(x)=xex and ₵(x)=exsinx+2∫0xcos(x-t)₵(t) dt

Q: 2. Evaluate the indefinite integrals: D = tan r secʻ r dr 8x + 7 dr 72 +1 E = - 2dr F = dr V9 - 36x2

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Q: H.W: 1- Evaluate the following integrals St7t?+12 dt - S- ol X (4x2 +1)3 a -

A: a.  ∫t7t2+12dtLet 7t2+12=x⇒14tdt=dx⇒tdt=dx14∴∫t7t2+12dt=114∫xdx=214×3x32+c=1217t2+1232+cwhere c is…

Q: Evaluate the integral: Se³+ A B C 3+cosh2x33+cosh2x sinh2x dx (3e)3+cosh2x + C 2+2ln3 (3e)3+ sinh2x…

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Q: 3. Evaluate the integrals. dx ( Use u = Vx(1+x)

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Q: Evaluate the indefinite integral ∫⟨1t,4t3,√t⟩dt, where t&gt;0.

A: We will integrate term by term.

Q: Q1)Evaluate the following integral: 2 1-x dx 1+x2,

A: 1 We have to evaluate the integral ∫021−x1+x22dx. Now,…

Q: Q2/Evaluated the integrals v2 1- f– (1-v2 )2

A: The integration of v2/(1-v2)5/2.

Q: 2. Evaluate these indefinite integrals: dt (d) S; dt 2t2+5t+3

A: Given that a integration  and we solve it by using general property of integration. Here i attached…

Q: 6. Evaluate the fallowing real integrals by residue. theerem 3 -bm -am 응) Co3 mX ans : X Sin mx ans…

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Q: dy J. 1. 2.

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Q: Consider the following. 9. dx x2 + 18 (a) Evaluate the integral with the Fundamental Theorem of…

A: The solution are next step

Q: 1 dz c+x2 c)

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Q: Evaluate the integrals

A: Our Aim is to evaluate the integral given below:-∫-π3π3 sin2(t) dt.

Q: i +j- t3/2 k ) dt |

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Q: Q1 - Evaluate the following integral: J (2x-3x+2) dx (x-1)2 (x-2)

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Q: Evaluate the integral d.x (V4 – x²)³ 0. -

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Q: 3. Find following integrals using method of substitution: Sze z dz a. 2 x2 b. S /1+x

A: Consider the provided integral, ∫ze-z22dz Find the expression for this integral using substitution.…

Q: (3) Using Hilbert-Schmidt theorem, solve the following integral equation: (1) y(=) = (x+2)*+ /…

A: Given integral equation, yx=x+22+∫-11xt+x2t2ytdt          ...1 Here fx=x+22, λ=1 Firstly find the…

Q: Evaluate the integral.   The integral from 1 to 4 of e^(4/x)/x^2dx

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Q: a) Compute these two integrals shown below using one application of trapezoidal rule and Simpson's…

A: Given that ∫23x dx ∫-66x2cosx dx we have to calculate given integral using  trapezoidal rule.

Q: 2.2 Find the indefinite integral of the following: I = S(4x² + 5x – 7)dx а. | b. I = S Vx dx I = J…

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Q: Evaluate the integral of ax^2 e^bx dx assume a and b are constants.

A: Please see the explanation below