1 lim x→0 x sin x x2 О 14 O 1/7 О 16 O 1/5

Q: - i) By using L'Hopital Rule, find sinx lim x-0 e3x 1 ii) Given that a = sin V3 2, find, csca -1

A: In L Hopital's rule take the derivative of numerator and denominator separately and than put the…

Q: Find the maximum and minimum valucs of f(x) = sin x + cos x on (0, 2n].

A: Given function:                      f(x) = sin x + cos x,   x∈[0, 2π] Use the second derivative…

Q: In (**) 1+v1-x²' sech-1 x (A) O sech x

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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: .Find the estimate for the root of f(x) = sin x + cos (x² + 1) – 1using Bisection methodx, = 1. 5,…

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Q: 4. (a) (b) Approximate the integral f x² cos xdx with n = 6 using the composite trapezoid rule;…

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Q: Perform error analysis of approximation derived (a) 1 x² 1 - at xo = 0 X (b) sin (₁) ₁ X- at x = = 0…

A: Given that: a 1x2-1 at x0=0b sin xx-1 at x=0c exp x2-1 at x=1 and x=-1 The objective is to analyse…

Q: 8 cos" 2πt dt

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Q: Evaluate the definite integral: r1 arctan x dx = x2 +1 0.

A: To solve the integral first I have make a substitution so that integral become easy and the solved…

Q: Use the Mean Value Theorem to show that f(r) = 2x sinh(r – 1) +2" – r has a critical number between…

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Q: Find lim t→ 1 1 - cos2 Int 0(0.20 를 를) 0 O t ./t+7, t-1 t2 − 1

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Q: Express lim,00 E= cos( + ) in integral notation. i=1 _2n

A: Since, the above equation is given in the form of Riemann's integral, which is given by-…

Q: O Approximate So In (3 + sin x)dx using Simpson's Rule with(n = 4 %3D

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Q: Q1- Prove that coth x = 2 csch 2x + tanh x

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Q: 4 1 : Evaluate the following improper integral by using proper limit notations: dx. (4 – x)2/3

A: As per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: Q1- Prove that coth x = 2 csch 2x + tanh x %3D

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Q: Prove that for all r € [1, 00), – sinh (- |cosh-(x)|) = vVI² – 1.

A: We know that: Cosh-1x=logex+x2-1  and  Sinhx=ex-e-x2. And, Sinh-x=-Sinhx. Therefore,…

Q: 2. (a) Show that when x > 1, In(x) (This should be very short.) (b) Use your knowledge of integrals…

A: We have to show below given expression

Q: Evaluate the definite integral. 3 2esin ( z ) cos(x)dx 3

A: Given that: ∫0π32esinx.cosxdx

Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: Evaluate the definite integral. T/2 esin(x) cos(x)dx

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Q: Consider , xe¬**dx. What will be the appropriate solution and answer to this improper integral? A.…

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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: 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: .Find the estimate for the root of f(x) = sin x + cos (x² + 1) – 1using Bisection methodx, = 1.5, X…

A: Solution

Q: - Using Greens date $(x2 + y)dx bounding of the 0<x<1 and x' sysx.

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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: Find the most general antiderivative of: f(x) = 4√x − sec(x) tan(x) + (7/(x2+1))

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Q: find the limit as x approaches 0. 12sin(3x)/1-cos(3x) = as x approaches 0

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Q: 1 lim t→0+ (뉴 1 Vt2 +t,

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Q: Find the limits sin(3t) lim In(sin(3t)) i+ In(tan(4t))" j+tln(3t) A t→0 sin(4t)

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Q: Find the most general antiderivative of: f(x) = 4√x − sec(x) tan(x) + (7/x2 +1)

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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: Evaluate the integral: sinh 3x csch 3x dx coth 3x "+C 3 coth X +C 3 - coth x +C - 3 - coth 3x +C 3

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Q: Evaluate the following integral cln 2 cosh x dx. sinh² 0. x + 4 sinh x + 2

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Q: (b) Approximate the definite integral f"/² cos(x)dx by employing the Composed Midpoint rule with n =…

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Q: 4. Evaluate if they exist: (a) sinh x dx (b) lim , sinh x dx r+00

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Q: Q4: To evaluate the integral fcos' xdx using the reduction formula of cosinc function

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Q: Evaluate the integral 3. 2х cos(xy)dydx 2

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Q: Express the limit as a definite integral on the given interval. lim [9(x;)³ – 9x;14x, [2, 8] i = 1…

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Q: Find the indefinite integral ∫ [arctan(x/2)] / (4 + x2) dx

A: ∫arctanx24+x2dx Now put, arctanx2=t⇒11+x22·12=dtdx⇒11+x24·12=dtdx⇒44+x2·12=dtdx⇒24+x2=dtdx⇒14+x2…

Q: Use Simpson's 1/3 rule with (n₁ = 2), and Simpson's 3/8 rule with (n₂ = 3) to estimate tan-¹(2).…

A: We have to estimate the given function.

Q: Using Simpson's (1/3) rule with h = 1, evaluate the integral: 7 = f x² logx dx 3

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Q: 46) Estimate the minimum number of subintervals needed to approximate the integral T/2 6 sin x dx…

A: Find the minimum number of Subintervals.

Q: Prove that for all x € [1, 00), – sinh (- |cosh'(x)|) = V -1. %3D Ouest!

A: Proving of function  sinh(cos-1(x))

Q: Find the antiderivative of f(x) = point (0,3). O F(x) = tan¹x +1 O F(x) = 2 tan ¹x +1 -1 O F(x) =…

A: Given: fx=24x2+1 and x,Fx=0,3   As we know the expression for the antiderivative; Fx=∫fxdx And…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.