21. Solve the equation: a) 2 tanh-1x = In 3

Q: Separate tanh-1(x+ iy) into real and imaginary parts.

A:

Q: Show that u(x, t) = 4 tan-1 I-ut ev where v is constant, is solution of Uxz – Utt = sin(u)

A: To check the given solution satisfy the given partial differential equation, substitute the solution…

Q: Obtain the general solution to the equation dr +r tan 0 de sec 0.

A: Given first order differential equation is drdθ+rtanθ = secθ. We have to find general solution of…

Q: Ex 15: Prove that : d du tanh u = sech2 u dx dx Proof:

A:

Q: 2. A hemisphere of radus 12 nches and base up is full of water that runs out through an orifice of…

A: Dimensions of a hemisphere tank are: Radius R=12 inches or 0.3048 mHeight of tank H1=Radius of tank…

Q: 2 Solve this Di fferential equat ion by applying ( variation paramet ers) 4+y = y+y%3DSE c x

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Q: 2ーてスース+スニ。 A ind yh ans: スi:z )スてニリ人3:)

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Q: Show that coshr + sinh r = e" and cosh r - sinhr = e. Hence, deduce M MOUTMOUT cosh? a – sinh? r 1.…

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Q: Suppose a particle has velocity given by v(t)=sin(t). What is the net displacement of the particle…

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Q: 2, Sove the Following Riccati equâtion dy 2008?x- Sin?x +y? %3D 2 cosX

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Q: sin(nz) b) E: on (0, 2x] n-1 AcoAnA

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Q: b) By using appropriate substitution, solve the integral 7x dx. (1–4x)"

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Q: let Po)- tar'c) + tant (L). show that Plx)=C,where cis a constant and then Aind C

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Q: as Let ĝ +sinh (z?) - ji = 27 where w=f (x,y,2)- %3D Find 2 nd JW and au

A: In the given question we have to find the partial derivatives of with respect to x ,y and z. here we…

Q: By Cauchy's integral formula show that, Se" esin e -2n cos(cose) de = 2n; "esine sin(cose) de = 0 0,

A: Cauchy's integral formula : Let U be an open subset of the complex plane C, and suppose the closed…

Q: cos“ xsin xdx

A: The given integral is ∫0π2cos4xsin5x dx. We solve the given integral by using general formulas of…

Q: find: de So 2-cose using sediment theorem

A: Solution :-

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Q: use in tergration by parts to solve

A: Given,  ∫012xcosπxdx  We have to find the integral using integration by parts.

Q: a) By expanding sin (2x + x) and using double angle formulae, show that sin 3x = 3 sin x – 4 sin°x .…

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Q: 1 Proof (I): (sinhx)' = Vx? +1

A: For direct Proof:

Q: According to the method of integration by parts, we have that |(72)"e" dz, with n natural number, is…

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Q: 1. ò F 2 dt dt Ve' + 9

A: The integral function are 1. ∫etet+9dt   2.∫2t+4ttdt

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A: The double integrals can be solved by solving the inner integral first followed by the outer…

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Q: By the samc way, prove that g 7 sinw.k MUIS Z N-27 cesW. +1

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Q: What is the general solution of sin z sin ydx + cos rcosydy 0? %3D

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Q: 2. Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a…

A: Given data: The work done is 2 J. The change in length is (42-30)=12 cm=0.12 m The expression for…

Q: 3. If v = sin- (4) + tan-' (±), find the value of r + y. 4. If z= ear+by. f(ax – by), prove that b…

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Q: Te 2=tnx then prove thot dx 28 Snメー tty? 1+xて

A: We have to use proper calculus applications.

Q: The value of integral cos (1+x) d, n= 4 by Trapezoidal rule is (up to 4 decimal digit) Answer:

A: The trapezoidal rule can be used to calculate the integral numerically. The formula for the…

Q: If the work required to stretch a spring 0.25?? beyond its natural length is 12 ?? − ??, how much…

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Q: Select the definitions of sinhx, tanh a, and sech z. O e +e- e -e-z O e-e e +e-z O e -e e +e- O e +e…

A: The hyperbolic function of sine is expressed as-     sinh x=ex-e-x2 Thus, Option (C) is the correct…

Q: 2. Apply appropriate trigonometric substitution to compute the integral 1-4x +2 dx /2x

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Q: Ex 15: Prove that : d du a) tanh u = sech² u dx dx

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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: Suppose that 8 J of work is needed to stretch a spring from its natural length of 40 cm to a length…

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Q: As shown in Figure 1, the volume (V) of liquid in a spherical tank of radius (R) is related to the…

A: V=volume of the filled water in the tank m3R=radius of the spherical tank mh=depth of the water in…

Q: 7a6(cos +i Si ) IF USE DE MOIURES TheoREM To Find Z. 23

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Q: 22. Determine Sz sin? y cos y dy

A: Given that: ∫0π2sin2ycos6ydy

Q: 2* csch (2*) cosh(2*)-sinh(2*)

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Q: Use the addition formulas to deriv TT 31. cos(x - sin x 2

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Q: Find all solutions to cosh z = i sinh z.

A: To find all the solutions of the below equation. coshz=isinhz

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Q: z² = e² z*in [0, 2]

A: According to Newton's method, if we want to find the root of any continuous function let's say f(x),…

Q: Give exomple of o line ar equation s unknown

A: First we see the defination of linear equation then give the answer.

Q: 6. y 45

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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.