Different substitutions dr a. Show that sin(2r – 1) + C using either Vx - x u = 2x – 1 or u = x dx b. Show that 2 sin Vĩ + C using u = Vĩ. Vx - x² c. Prove the identity 2 sin¬lVĩ – sin(2x – 1) = (Source: The College Mathematics Journal, 32, 5, Nov 2001)

Q: 13. To evaluate f 3cos*x-1 dx. the integrand can be transformed to 1-sin2x a. [(3cos x+ csc²x)dx b.…

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A: Consider the homogeneous differential equation y'sin2x-ycos2x=0. Variable separable form:…

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A: AS per company guidelines we  are supposed to  answer only one question at a time. For solution of…

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A: Let's find.

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A: As per our policy, we are allowed to attempt only first question. Kindly post remaining question…

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A: Ok....to verify this, we will have to differentiate the RHS of the equation, if we get the function…

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A: See the attachment

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A: We have to solve above integral

Q: #2. Verify Clairaut's Theorem for the given function f (s, t)= In (st)-s* sin (6t) +st.

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A: ∵1     ∫0π4(1+etan θ) sec2θ dθ=∫0π4(sec2θ+etan θsec2θ )dθ=∫0π4sec2θ dθ +∫0π4etan θsec2θ dθ∵∫0π4sec2θ…

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A: You need to have the idea how to use substitution in integration.

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A: Given,∫π22π312+sinx-cosx

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Q: Consider ∫ x sinx cosx dx = f(x) sin2x - g(x) cos2x + C. What is the value of f(1) + g(1)?     3/8…

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Q: 4* What is the amplitude of f(x) 1 cos(TZ – 1) + 2? %3D 3 a. 2

A: Here we have to find the amplitude of f(x)=12cosπx-1+2

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A: Given that z =x+iy For a part , y<0 and we need to prove that | eiz | >1 For b part, we need…

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A: Given ,  u = f(x)

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Q: to the I. sinx(e+ 1)x' - cosx = 1, y(=) = 0 %3D %3D (1 + e) (1 + cos x) = 0 (1 + e) (1 + sin x) = 2…

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Q: - Using Euler's equation, show that- : z fain (K 0)} - z sin Ø z2 – 2zCos Ø +1

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Q: A. Ifw= sin(x+bt)+cos(2x+ 2bt) . Prove that ôx? B. If f = x*y* + 2y sinx + x² lny then show that fxy…

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

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Q: 1. Determine the relative extrema of the function y = 2 cos x + sin 2x on the interval (0, 2Tt). %3D

A: To find the extrema of the function

Q: 9. Integrate the following: a) S sin? x cosx dx b) "(1 – cos 2x)/2dx

A: (b) The given integration is: ∫0π1-cos2x32dx To find the value of integration.

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A: f(x) is a sin function with amplitute 1, and is such that f(0)=0 f(1)=0 f(x) &gt;= 0 for x element…

Q: Is there a value of b that will make f (x) = {cos a, r>0 differentiable at x = 0? Upload Choose a…

A: Explained below

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Q: Evaluate 3 sin(x) cos(x) dx by four methods. (a) the substitution u = cos(x) 0-흑 cos2(x) + C O…

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