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

Q: Evaluate the definite integral.

A: We have to evaluate the integration ∫0π3sec32xtanxdx

Q: Use Simpson's 1/3 rule with (n₁ = 2), and Simpson -2 1 estimate tan-¹(2). [Hint: tan-¹ (2) = ₁²; dxl

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

Q: V3 B) If sinh 0 Find cosh 0, tanh 0, coth 0. %3D 2

A: We are given with sinhθ=32. We need to find some other trigonometric functions. Use the relation…

Q: Q6:-Prove that: y" =1 y =csc(tan"") if

A: Given y=x4·csctan-114x2-1 Let tan-114x2-1=α ⇒tan(α)=14x2-1 ⇒cscα=2x ⇒y=x4cscα⇒y=x4(2x)⇒y=x22

Q: Show that y = tanh x is an odd function.

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Q: f(x)% 3D tanh In(2x) is equivalent to

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Q: 29.Verify that W[xλ cos(μ ln x), xλ sin(μ ln x)] = μx2λ − 1.

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Q: prove f sinh(x) cosh(x) dæ = } cosh(2x) + C" %3D

A: Evaluate the given integral as follows. ∫sinhxcoshxdx=12∫2sinhxcoshxdx=12∫sinh2xdx…

Q: 1. Determine the absolute extrema of f(x) = (cos x − 1) sinx on [0, π], if they exist, and identify…

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

Q: For x >0, tanh(Inx)+ dx x2 - (tan-(x)) 1 + > = x? .2 Select one: O True O False

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Q: find the critical points:

A: Given function is  and the interval isThe critical points of a function are obtained when the first…

Q: Use the addition formulas for sinh x and cosh x to prove tanh u + tanh v tanh(u + v) = 1+ tanh u…

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Q: Prove that sin x can be approximated by x − (x^3/6) within 0.01 on the interval [-1,1].

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Q: 2 1- cosh-1 x = In(x+Vx² –1 2- coth¬lx = x+1 In 2 x-1 Vi+x? 1 3- csch hx= ln(÷+

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Q: hich the extrema occur. g(x)=x^2+cos⁡(x^2 ) on [-√π,√π]

A: The function given is gx=x2+cosx2 on the interval [-√π,√π]. To find the extrema of the function…

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

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Q: Find y' if y=In(tanhx) +cosh¬1(coth2x)

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Q: Q3: Let cosh(x)3D5, x>0 , Find values of: sinh(x), tanh(x).coth x ,sech x,csch x

A: Given, cosh(x)=5, x>0

Q: 2 sin x f(x) = 1+2cos x Pvaluated at x = 1 ran

A: Given: fx=2sinx1+2cosx

Q: +Jx² +1) Sinh'x=ln(x+ 2. 1. Cosh-x = In(x+

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Q: Show that O tanh(In x) = x² -1 O (cosh x + sinh x)" = cosh nx + sinh nx %3D

A: From the given problem:   tanhlnx=x2-1x2+1   As we know that :   sinhx=ex-e-x2 ; coshx=ex+e-x2   So,…

Q: None Q2) prove that cosh(x + y) = coshx coshy + sinhx sinhy

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Q: d 4. Prove that dr (csc (r)) = - csc (x) cot (x). CSC

A: We have to proof

Q: Find y' if y= In(tanhx) +cosh-(coth2x) Select one: O y' = cothx- 2csch2x 1 O y = -2csch2x sinh x…

A: The differential of products of trigonometric functions can be calculated using the product rule.…

Q: 9. Differentiate y = cos(x3/2 In x), do not simplify. x2-4 Using Calculus, show on which intervals…

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Q: Show that O tanh(In x) = (cosh x + sinh x)" = cosh nx +sinh nx

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Q: 3 tanh (In(4x)) sech" (In(4x)), I) y = tanh (In(4.x)) [y

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Q: 9. Solve tan x for x where 0<x<90* V3 Enter your next step here

A: Just observe that its value of tan(30°).

Q: hD (Hint: E =eD and Sinh x: 2 Q3: Prove that 8 = 2Sinh %3D

A: We will prove the following result.

Q: B) If sinh 0 V3 Find cosh 0, tanh 0, coth 0.

A: Given: sinhθ=32 To find: coshθ, tanhθ, cothθ. Solution: We have identity, cosh2θ-sinh2θ=1 Putting…

Q: 1. Determine the absolute extrema of f(x) = (sin x – 1) cos x on 2' 2 if they exist, and identify…

A: Given:-f(x)=sinx-1cosx,        x∈-π2, π2f'(x)=sinx-1ddxcosx+cosxddxsinx-1…

Q: Hico/ prove Thall Cosh2: - 2 tañ z Sinhz: 1+2 Tanhiz =

A: Since you have asked for a question with multiple sub-parts , we will solve the first three…

Q: 1-sin x F(x) = S₂₁² 5tdt

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Q: 1 Use the definition of the hyperbolic functions to prove the identity 1– - = tanh?(2x) cosh?(2x)

A: Explanation of the answer is as follows

Q: 7. Determine f (x) so that | cos(In°x) f (x) dx = sin(In²x) + C 2 In x In x (а) 2 In x (c) - (d)?

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Q: Find F' if rsin ) = /" 4 nin" t t. F(x) = cos-lr

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Q: Evaluate integral 0 to pie sin 2 x cos 4 x dx

A: To Evaluate the integral given below:-I=∫0πsin2(x)cos4(x)dx

Q: 1. Evaluate f{(t² e2t + sinh2t)}

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

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

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Q: 1 Use the definition of the hyperbolic functions to prove the identity 1- - tanh? (2x) cosh (2x)

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Q: Q6:-Prove that: y" = 1 y =csc(tan"" ) if

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Q: (a) Use the definitions above to show that d tanh x dr 1 cosh r

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Q: Find all values of x on the interval L0,27) that ssue the following equabion coslex) +sin(n) -l=0

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Q: a. Find the root mean square value of tanh x, for 0 < x< 2.

A: we know the formula of root mean square, rms = [ (1/b-a) (integration of |f(x)|2 from a to b) ](1/2)…

Q: 1. cosh (j) 2. tanh 3. 3. In(2 + j3)

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Q: HiW Q Prove that : tanhclax) = x-! %3D x2+1

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Q: prove using the IVT. e^x + ln x = 0 has a solution.

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

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

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.