1. (a) Use integration by parts to verify the reduction formula Scos" xdx = cos"- xsin x n-1 +- Scos"-² xdx n n Use this result to derive the Wallis's cosine formulas (i) If n is odd (n23), then 4 6 =| 3 | cos" xdx (ii) If n is even (n> 2),then a/2 -1) T 3 | cos" xdx =| 2) 4 6 2 n

Q: ди 4. Solve = a given that (i) u = 0 when x = 0 and x = l for all t at (ii) u = 3 sin TT X when t =…

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Q: 7.Obtain the general solution using the method of undetermined coefficients. a. (D² + D)y = - cos x…

A: First we put auxiliary equation D2+D=0 then solve it.

Q: 38. Solve the integral / x³ /x² – 1 dx three ways: (a) with the substitution u = x? –- 1, followed…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: The integral of the given function | cos(9x) cos(5x)dx is equal .. 14 sin 4x + 4 sin 14x А. · + C 12…

A: This question can be solved by applying the formula of 2.cosA.cosB

Q: 4.Find the temperature T (x, t) in an infinite bar if the initial temperature is 4 for |x|3. Solve…

A: Sol

Q: 2- Find the positive root of f(x) = xe" - cos(x) with an error tolerance g=0.005% by using %3D a)…

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Q: 1 = 1 +x+x?+ 1-x Ex", I지<1 n=0 1 = 1-x+x2- ... + (-x)"+ 1+x E(-1)"x", x|<1 n=0 x2 ex = 1+x+ 2! x"…

A: use e^x expension

Q: Step 3 Substitute cos 60 = u. Differentiate with respect to 8 on both sides. -6 sin 60 de = du Step…

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Q: (a) Find out the definite Integral, From 0.5 to 1.0 of Cos(x) dx.                           1.0…

A: (a) Evaluate the given integral as follows. ∫0.51cos x dx=sin…

Q: 21.1 Evaluate the following integral: * 7/2 (6 + 3 cos x) dx (a) analytically; (b) single…

A: Solution :-

Q: 2. Solve for y in terms of x for the following equations: a) In(1- 2y) = x %3D b) In(y- 1) - In 2 =…

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Q: 4. Let f(x) x² + cos(Tr). f&)= Qx -Tsin(Tx) (a) Use Newton's method to find the root of f(x) with x1…

A: a) Given f(x)=x2+cosπx  with x1=-12=-0.5    f'(x)=2x-πsinπx  we know Newton's formula to find…

Q: 2 /2 L(x)=- I Jo cosh(x sin 0)d0. Use the composite Simpson's 1/3 rule with step length h = T1/16 to…

A: Use the composite Simpson's 1/3 rule ∫abf(x)dx≈∆x3fx0+4fx1+2fx2+4fx3+  ...   4fxn-1+f(xn) here…

Q: 41. Let a, b, r bc constants. Show that k sin rt - r cos rt y = Ce-kt +a + bk k2 + r2 is a general…

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Q: 2- Exercise: Evaluate a) L{4x2 – 3 cos(2x) +5e*} b) L{4+ sin (x) – e** } -

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Q: 2. 1 Evaluate the definite integrals below. Use u-substitution when appropriate. 4 5 3el/x (a) (b) |…

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Q: 1. (a) Use integration by parts to verify the reduction formula cos" x sin x n Scos" xdr = -1 [cos"…

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Q: 2- The integration of f x²sin (x³). dx is: (a) cos (x³) + C (b) cos (x³) (e)금 cos (x3) + C (d) 글 cos…

A: According to the bartleby guidelines, for more tan one question asked only the first question should…

Q: Find the Wronskian of the functions ƒ(0) = cos²(0) and g(0) = 1 + cos(20). NOTE: Your answer must be…

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Q: 1. a) For 0<x<1, show that /2 1+ x b) Use the estimate in a) and integrate it to prove that 1 1 -dx…

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Q: Solve the given initial-value problem in which the input function g(x) is discontinuous. [Hint:…

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Q: /2 1. (a) Use the Wallis cosine formula to evaluate cos' x dx. (b) Evaluate | 1 - dx by using…

A: Wallis's formula state that 1. if  n is odd and n≥3,then ∫0π2cosnxdx=23.45.67 . . . n-1n Therefore,…

Q: 1 (27 x sin nxdx 27 bE f(x)sin nxdx -sin nx -1. 1-2n cos 2nn COs nx %3D %3D The requirod f 2)

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Q: 1/9 Determine the trapezoidal approximation of 7 sin 9x dx using 4 equal subdivisions of the…

A: Given integral is ∫0π97sin(9x)dx. To find the value of this integral by trapezoidal rule by taking 4…

Q: 4- A possible form for a particular solution of y"+2y = cos(v2 t) is a) Acos(2 t) + B sin(/21) b)…

A: This question is related to differential equations, we will solve it using given information.

Q: 4. Evaluate the iterated integral T, cos(³) dydx. a. (cos 1- 1) b. 1 e. (sin 1- 1) f. 1 (1- cos 1)…

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Q: 1 (a) Use the trigonometric substitution x=5 tan 0 to calculate the indefinite integral dx. Show…

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Q: Which of the following is one of the correct steps to evaluate cos'x dx to obtain the desired…

A: Solving

Q: 5 to x = 19 using 9 rectangles In order to approximate the area under f(x) = cos(9x) + 7 from x =…

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Q: 18. What will be the value (if any) of the particular solution of the problem u (1+ cost)u=1+ cos t,…

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Q: 1. Find the particular solution to y' = 3sin(x) given the general solution is y = C - 3cos(x) and…

A: Topic - differential equations

Q: 9 Solve the equation for O< x < 2TT -5 = 1+3 cos x 10. solve the eycharm for X over 0< x < 2 το 2…

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: 1- The integration of ſ cos . dx is: (a) sin를 + C (b) 2 sin +C coS x (c) 2 cos를+C (d) + C 2

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Q: 4. (a) Consider the ODE y' + tan(ax)y = sin(ax), where a is a nonzero real number. Find an…

A: We will use basic knowledge of ODEs to solve this question.

Q: () 13. У 3к = 4", Vn E N k k=0 14. L 2k = 2"-1, where n is even k=0 15. У 4* = 5", Vn E N k k=0 SIN

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Q: 9 In order to evaluate the following integral 1 -√2/cos-¹(√x) dx 1 =- we first use the substitution…

A: Given: I=-∫12xcos-1(x)dx

Q: . (a) Write cos 3x + i sin 3x = e3ix = (cos x + i sin x)³ by Euler's formula, expand, and equate…

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Q: [9] Find a 27-periodic solution u(x) of 5u"(x) + 3u(x) = f(x) where f(x) is a 27-periodic function…

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Q: 1. (a) Use integration by parts to verify the reduction formula l(cos"-² x dx cos"- xsin x, n–1…

A: Solution:Verify the reduction formula∫cosnx dx=cosn-1x sin xn+n-1n∫cosn-2 x dxDerive the Wallis…

Q: b- Compute the roof of the function f(x) = cos(x) +2 sin(x) + x². Please use xo = 0 and x1 = -0.1 as…

A: 4) Instead of x1,x2,x3 I choose variables x,y,z a)

Q: 1- The area under the curve y = sin?x over the interval [0 to 2n] is: A. В. п C. 2- The value of…

A: Explanation of the answer is as follows

Q: J (sin 3x)² = sin 3x xp- sin 3x CO = sin 3x xp- sin 3x On simplifying further, y2 (x) = cq sin 3x /…

A: In this question , constant is to be included in the last solution of the differential equation.…

Q: Find a formula involving integrals for a particular solution of  D.E y^‴-3y^″-y^'+3y=sect

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Q: 9. Using integration by parts, Then k is? (a) - 1.5136 [*²x³ cos(x²) dx = k − [*³x sin(x²) dx -…

A: Given integral is ∫02x3cosx2dx

Q: find the particular solution to the general solution y(x) = c1x +c2+ x^2 -1 if y(1) = 1, y’ (1) = 2…

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Q: 16. So the maximum occurs at an angle of 0, but only if the angle is small enough. Explain why that…

A: Given: v=2μgscosθ+μsinθ Here μ is the coefficient of friction, g is the acceleration due to gravity…

Q: 36 - With p0=0.5 and p1=pi/4 starting point f(x)=cosx-x, find its root when n=0 using Secant Method.…

A: Given: To find the root of the following equation using secant method, f(x)=cos(x)-x Initial points:…

Q: 2 tan (4x – 1) – 3 cos?(4x – 1) cos (4x – 1) If we let u= 4x -1 for the integral of dx, what will be…

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Q: 2. By using Simpson's rule with N = 4,8, and 16 Integrate the following functions SOLVE PART B eX…

A: We need to use Simpson's rule to evaluate the integral. So, we need to find ∫0π4Tan X dX If we need…

Q: The integral of the given function cos 2x cos(-x)dx is equal . sin 3x + 3 sinx А. + C 6 sin 3x – 3…

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