@Hiw/ prove that cosiz=coshz @ Cosh'z -sinh'z=1 @1-Tanhz Sechz D cos hiz-l = cschz COS 3 =- Sinh z sinh (-2) Cosh(-2)=cosha

Q: IT Evaluate the integral: (cos x + sinx cosx)dx

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Q: sin z In(sec z) dr

A: Answer and explanation is given below.. For 17 and 18 please repost the questions

Q: Show that cosh t + sinht = e and cosht - sinht = e. Hence, show that cosht- sinht=1. Deduce that…

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Q: If tanx=sinha prove that a=ln(secx+tanx

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Q: solve using L'Hopital's Rule 1. lim      [ 1/(sin^2 x) ]  - [ sinx/x^2 ]    x->0

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Q: Find the indefinite integral.

A: According to power rule of integration .

Q: 2x[ cos (3x) – 1] sin (6x) – 6x Use L'Hopital's Rule to evaluate lim 2x[cos (3x) – 1] lim sin (6x) –…

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Q: | 35. Prove the reduction formula for n > 2: cosh" x dx = cosh"- x sinh x + cosh" п-2 хdx п n-1 п п

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Q: alue Theorem to approximate the value of

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Q: In estimating cos(3x)dx using Trapezoidal and Simpson's rule with n = 8, we can estimate the error 1…

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Q: 4. Determine exact values of 0 if 0 < 0s 2n given that tano - V3. Explain your process.

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Q: integral of 1/sin3x dx

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Q: Prove: cosh 3x = 4cosh'x- 3 cosh x %3D

A: We will solve the following.

Q: can you prove it, please ? cosh 2x=sinh^2x+1 = 2 cosh^2-1

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Q: Verify the identity sinh x + cosh x = ex

A: To prove the identity

Q: 3x² dx

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Q: 15. sinh 2x = 2 sinh x cosh x %3D

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Q: (sinh x + cosh х)*3 еx

A: Solution:-

Q: Given: 4 2 y = -arctan sinh 3 х — find y' at x = 2 - cosh 1 - y 1 + sinh? 1 %3D cosh 1 y = 1 + sinh…

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Q: Prove the identity. cosh(x) + sinh(x) = e* cosh(x) + sinh(x)

A: The solution of the problem is given in the next step.

Q: 1. f(cos 3x - x²) (sin 3x- x³)5 dx 2. f cos² x (tan x) (sec² x − 1) dx 3. f tan x (sec² x - 1)(sec²…

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

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Q: 4. When a 2-kg block is suspended from a spring, the spring is stretched a distance of 40 mm.…

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Q: Hiw/ It isinh z = Siniz, prove That: sinhz = sinhxcosy tising cosh x

A: Given - If i sinhz = siniz To Prove - Prove that sinhz = sinhx cosy + i siny coshx         Formula…

Q: (2x³ + 5x-4) dx

A: Solved below.

Q: (5) sinh 2x = 2.sinh x.coshx %3D

A: (5)To prove the following identity sinh(2x) = 2·sinh(x)·cosh(x)

Q: integral of sin2x.cos2x dx

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Q: TRC Prove that CoshZ =Goshx Cosytrisinh X Siny

A: The given problem is to prove the following complex hyperbolic trigonometric identity.

Q: If tanh(x) = 24, find the values of the other hyperbolic functions at x. sinh(x) %3D cosh(x) coth(x)…

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Q: Limit as x approaches 0 of Sin(5x)/(2x

A: Limit as x approaches 0 of Sin(5x)/(2x

Q: Prove the addition formula for cosh x given by cosh(x + y) = cosh x cosh y + sinh x sinh y.

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Q: f(x) = = x, over the interval - < x < and has period 27 f(x) [cos cosz+cos3z + cos 5 +... f(x)…

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Q: Show that arctan(tanh(x)) = sech(2x). arctan(tanh(x)) dx D²tanh(x)) %3D 1 + ])-: 1/cosh²(x) 1 +…

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Q: Find the equation of the tangent line to the astroid x = cos° (0), y = sin³ (0) at 0 = 5. (Express…

A: The given equation of the astroid:x=cos3θ and y=sin3θWe need to find the tangent line to the astroid…

Q: Prove that: sinhx + coshx = e* %3D

A: we have  sinh(x)=ex-e-x2cosh(x)=ex+e-x2

Q: prove that cosh^2x-sinh^2x=1

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

Q: 3) Find the expention of sin³ 0 4) Proof that: cosh iz = cos z 5) Prove that: cosh² z - sinh²z = 1…

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Q: solve using L'Hopital's Rule 2. Lim      (x-sinx) / (x^3)    x->0

A: To find: The limit of limx→0x-sin xx3. Calculation: Apply L'Hospital rule: limx→0x-sin…

Q: 2 sin'x - cos x sin'x - (1 - sin'x) in*s) Rule ? %3D sin°x - 1 + sin´x Rule ? %3D = 2 sinx - 1 Rule…

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Q: Prove that cosh(x±y) = cosh x cosh y± sinh x sinh y.

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Q: Prove the identity (cosh x + sinh x) n  = cosh n  x + sinh nx

A: Consider the given L.H.S of the given identity coshx+sinhhxn

Q: prove that cosh(x + y) = coshx coshy + sinhx sinhy

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Q: solve using L'Hopital's Rule 3.  Lim    (ex + e-x  - x2 - 2) / (sin2x - x2)    x->0

A: L'Hopital's Rule: The rule is applicable when the value of the limit function limx→af(x)g(x) is…

Q: tanx(1+cot^(2)x)=(1)/(cosxsinx

A: To prove tanx1+cot2x=1cosxsinx consider the left hand side  tanx1+cot2x=1cosxsinx we know that…

Q: 2) tan-l 1 Vz 4) sin-! 6) tan-1V3 8) tan-l 0 V3 10) sin-! V3 12) tan- 2. 2. 3.

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Q: sin³ydy 1+cosy

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Q: sin- In2| 4+ tan¬'cos- dx 2 2 1+ cos- |4+ tan-'cos= 2 2sin- In2| 4+ tan-'cos- |dx 2 2 A 1+ cos2…

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Q: Find 2 sin(x + sin(x + sin a)) – 2 cos(cos 1) lim 2x – T ithout using L'Hôpital's Rule.

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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.