17. Use the theory of residue to evaluate the following definite integrals 2 de (2) / 2+ cos 0' cos 30do 5 — 4 сos 0' (b) cos 20d0 (c) T where (-1 < а < 1), 1 — 2а сos 0 +а2 - sin2" Ode where n = п %3D 1, 2, .... Ans. (a) 27//3, (b) 7/12, (c) na²/(1 – a²), (d) a (2n)!/ (22" (n!)²).

Q: Q4) Solve the integral by Simpson's rule x3 cos2x dx, h 0.2

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Q: Evaluate the integral. 3 1 3y2 cos xy dy dx x/3 5(1 - cos 3) 3(1 - cos 3) cos 3 - 1 (1 - cos 3)

A: Formulae used: ddxxn=nxn-1∫cos(ax)dx=sin(ax)a+C∫sin(x)dx=-cos(x)+Csin(0)=0cos(0)=1

Q: 18 Estimate the error if cos t is approximated by 1- 2 in the integral cos t? dt. 4! - -+ The error…

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Q: 8. Find the general indefinite integral (x³ -2² + x)dr; b) / (2" +-)dr, x > 0; c) (sin(3t) +…

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Q: According to numerical methods, the integral of a function f(x) over the limits a and b can be…

A: Given data  Used Simpsons 1/3 rule  f(x)=7e^(-πx)(sin(x^2)) With a=0 and b=2 and number of intervals…

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. S 20x cos x dx Select one: O A.80 sin. 1 40x cos + C O B.20 sin n1)x+ 40x cos…

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Q: Evaluate the integral: Se3+ cosh2x33+ cosh2xsinh2x dx A (3e)3+ sinh2x 21n3 - + C (3e)3 + cosh2x B…

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Q: 7. Express the solution of y" + 2y' + 2y = sin (3t) convolution integral (you don't need to evaluate…

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Q: Find the indefinite integral. |cos 2x ein 2* de sin 2x A) 2 +C ein 2x +C B) C) sin 2x e cos2x +C D)…

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Q: If xy In(3 sinh( 3x)), then x2y'= %3D Select one: Ox coth( 3x ) - In( 3 sinh(3x)) 09x coth( 3x ) - 3…

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Q: 5 Evaluate the integral 2x cos(xy)dydx 4 0.

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Q: Estimate the error if cos t² is approximated by 1 integral , cos r² dt. 4! in the 2

A: We have,

Q: x³ + 6x2 + x+2 2 Tan Ve fx-1 ax + In{(x+ 1)*(x² – x + 1)"| + The resulting integral of -X+ -1 + C,…

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Q: 13. Evaluate the indefinite integral fx cos x dx x sin 2x cos 2x a. + c 2 x sin 2x cos 4x b. + c 2 2…

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Q: 3. Evaluate the double integral /sin(x+y) dx dy by using Simpson's 1/3-rule with four sub-intervals.

A: I = ∫0π2∫0π2sin x+ydxdy,

Q: 3. Prove properties (3) and (4) analytically, by making x = -t in the part of the Integral from -a…

A: Concept:

Q: In the integration of (cos y)^2 dy/(1 - sin y), what is the first step to solve the integral? a.…

A: Consider the integral: ∫cosy21-sinydy

Q: 1. Evaluate the integral 2 1-2 8x cos y dy dr y - 4 by reversing the order of integration.

A:

Q: Find the indefinite integral ∫cos 2x cos 6x dx

A: Given integration as I=∫cos 2x cos 6x dx Since  2cos 2x cos 6x=cos 6x+2x+cos 6x-2x…

Q: Q2:- Apply romberg's method to the integral of f(x) = nx*cos(-Tx to 2 with Tol = 10-4

A: Given function is f(x)=14πx4cos14πx, we have to integrate above function from 0 to 2. Formula for…

Q: Compute the definite integral T/4 0 5 sin² (3x) dx

A: Sol:- ∫0π45sin23xdxTake the constant out:  ∫a·fxdx=a·∫fxdx=5·∫0π4sin23xdx=5·∫0π4121-cos6xdxTake the…

Q: 12.)Express lim E(3(x;,)² – 5x, + 2)Ax as a definite integral on the interval [0,1]. i=1

A: The given expression is: limn→∞ ∑i=1n 3xi2-5xi+2∆x    on interval [0, 1]   The definite integral  on…

Q: The integral of the given function | sin 9x cos 4x dx is equal. .... 13 cos(5x) — 5 сos(13x) А. + C…

A:

Q: please explain how to do question 7c

A: The given integral is ∫(1-3x)4dx.

Q: Eliminate the arbitrary constants 1. X = -Int, y= at + bt2 2. In y = C, sin ax + C, cos ax + 9x<-2…

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Q: Evaluate the integral. 3y- cos xy dy dx x/3 O 5(1 - cos 3) O 3(1 - cos 3) O cos 3 -1 (1 cos 3)

A: Given: ∫03∫x313y2 cos xy dy dx We have x3≤y≤1 and 0≤x≤3 This can be seen as: We can change the…

Q: 8 (a) To evaluate the integral J cos xdx by using the reduction formula of Cosine function.

A: We have to use cosine reduction formula.

Q: 5e3* dx J 6. 9-12e3 +4e6x

A: integration

Q: 3+ Evaluate the integral: Se³+ cosh2x33+ cosh2xsinh2x dx (3e)³ 3+ sinh2x A + C 21n3 (3e)3 + sinh2x B…

A: Application of integration

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: Q4: a)Evaluate the integral: ( In(cosh2x) coth2x dx b) prove that sec-1x = cos-1! = Cos

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Q: 11. In line integrals, if x(t) has a derivative equal to 0, with y'(t) and z' (t) not equal to 0, A.…

A: To Select: The correct the option(s): 11. In line integrals, If x(t) has a derivative equal to 0 and…

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: 5 Only one of these calculations is correct to evaluate the definite integral 7x In(x) dx Using the…

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Q: 2π 3 [³* sin(y) + sec² (y) dy

A: We have to evaluate the given integral.

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

A: ∫ 3x lnx dx 3∫ 1lnx·1x dx

Q: sin ® 3y cos3ydy cos 3y dy

A: We have to find out the integral

Q: 11. In line integrals, if x(t) has a derivative equal to 0, with y'(t) and z'(t) not equal to 0, A.…

A:

Q: You found that the reduction formula of a certain integral is In = a" sin 2x + nx"-1 cos 2x – n (n –…

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Q: - Prove using integrals Proper tos : e even sin()fc) dx=0 -b 9- For even functions.

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Q: 14. Use a suitable change of variables to determine the indefinite integral. fsin²xcos³ xdx (Hint:…

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

A: Use the following formulae, to evaluate the given complex integral. Let z0 be a pole of order n,…

Q: You found that the reduction formula of a certain integral is In = 180 (13x" sin (13x) + næ"-1 cos…

A:

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: Select the improper integrals in the following list of definite integrals. 2 7dt o 1-t2 1 9x In(x)dx…

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Q: (integral of -4 to 4) of 5*cos(3t)(delta(t))dt

A: Here,

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: Find a formula involving integrals for a particular solution of D.E y" + 3y" – y' – 3 y = tan x

A: In this question, we find the particular solution of the differential equation. Here, we used the…

Q: 1) Find the indefinite integral S(3x* – 6vx + 3 cos x – 4e*)dx. 2) Find S 4(sin(6 sin x + x² + Vx))…

A: