From the definition (Equation 4.46), construct »¡ (x) and n2(x). 1 d je(x) = (-x)' 1 d ; ne(x) = - (–x)' r dx sin x cOs x x dx (4.46)

Q: (b) Prove that : - 1) (coshx)2 – (sinhx)2 = 1 ,2) coshx- sinhx = e-*, 3) (cothx)? -1= (cschx)?

A: Given that  The following

Q: Solve 2 sinh(x)– cosh(x)= 5 by using the definition of the hyperbolic functions.

A: We know thatsinhθ=eθ-e-θ2coshθ=eθ+e-θ2

Q: Find the derivative of O2 sec 2x O 2tan 2x O-2 cot2x O 1/cos2x y=In sin 2x

A:

Q: On (-∞, ∞), three functions y₁(x) = cos 2x, y2(x) = sin² x, y3(x) = cos²x are linearly independent…

A:

Q: |2tan x In (cos x) dx=

A:

Q: Solve Cauchy Euler DE 4x^2y"-4xy'+3y=sin ln(-x), x less than 0

A: Let t =-x >0. Then the differential equation       4x2y''-4xy'+3y=sinln-x becomes…

Q: find the length of the curve y = arcsin e^(-x) from x = 0 to x = ln 5/4

A: Given curve, f(x)=arcsin e-x from x=0 to x=ln54

Q: Integrate: S tan (æ) sec³(x)dx

A:

Q: Express in exact form: In(x² + 1)dx

A:

Q: 2tan x In (cos x) dx=

A: here we have to integrate the given integral.

Q: 1- Prove that cosh 2x - sinh 2x = e2

A: sol)Given cos h 2x-sin h 2x=e2x

Q: By using the definition of hyperbolic functions, show that sinh x = 1 (cosh 2x – 1). -

A: Given,         sinh2x=12cosh2x-1

Q: What is the integral of question 23 and why is the answer 0?

A: Let f(t)= t^4*tan t/(2+cos t)

Q: 2. Evaluate the integral: | sinh x cosh² x dx

A:

Q: If y- sinhx then prove that = dx

A: Fundamental inverse Trigonometric Identities: →cosh2y - sinh2y = 1 Basic derivative:…

Q: limit x to 0 sin(4t)sec(1000t)/10t

A:

Q: If y = √√1-x² sin¹¹x, prove that (1-x²)+xy=1-x². dx

A:

Q: Evaluate S sinh x cosh x dx by letting u=cosh x

A:

Q: If Z= sinh () show that: *=+y==0

A: GivenZ=sinhyx

Q: )if x+tany+ sin-x+ In(yx) +3**1 =4 then find y'

A:

Q: (27 (b) Show that 0. 4 tanh- 2x dx In 8. 16

A:

Q: Find the first derivative of y = arctan [ (3 sin x) / (4 + 5 cos x) ] and y = ( ex – e –x ) / (ex…

A:

Q: (b) Prove that : - 1) (coshx)² – (sinhx)² = 1

A:

Q: Consider sin¹x cos³x dx. If we let u= sinx, the integrand can be transformed to

A:

Q: If xy-In(3 sinh(3x)) , then x? y'-

A: Given,           xy=ln3sinh3x

Q: 1+ tanx y = logy 1- tanx prove that 9 = sec2x dy dx

A: Given,           y=log1+tanx1-tanx

Q: 2.Find , if y = sinh x %3D dx' 1+cosh x

A:

Q: Compute sin x dx er + 3e -x 8.

A: We have to compute the given expression ∫sinxex+3e-xdx from -∞ to ∞ Therefore, The  given expression…

Q: 3. J sin53x dx. 4. J tan 2x sec22x dx.

A: ∫ax+bn dx=1aax+bn+1n+1+C

Q: rIn 2 cosh x dx. Vsinh a x + 4 sinh x +2

A:

Q: If Z= sinh () x- show that:

A:

Q: 8. The value of f sinh* x cosh³ x dx

A:

Q: From 0-2 sin(x^2) dx using trapezoidal rule with n=4 as well as midpoint rule with n=4

A: Trapezoidal rule=∫02sin2(x)......(1)As we know that accoding to trapezoidal rule…

Q: Solve x - cos(x) = 0 by Secant method. Use x-1 = 0.5 and xo = 1.0 as the initial estimates. %3D

A: We will find roots till 4 digit repeat

Q: Va2-x² Y=Arc sin + а

A: Differentiate wrt x

Q: 3. Prove that: cosh x - sinh x = e¯*

A: Proved Coshx - Sinhx = e-x

Q: 4. Ify =/sin x+ /sin x+.to inf inity, dy _ cosx dx 2y-1 prove that

A: Derivative of Sinx is Cosx

Q: Find* esin ¤ dx tan x csc x cos x e $in xt. - cos x e Sin x + c Option 4 Option 2 esinx + c ,sin.…

A:

Q: Su+1 dn th-2

A: I =   ∫135x + 1x2 + x - 2dx    = ∫135x + 1x2 + x - 2dxWriting5x + 1  = A (2x + 1) +BOn solving , A…

Q: aluc of 4 alue of dx. 1+(x – 2)? 1

A: See the details solution in below

Q: ind the equation of the line that is tangent to y = 2x + sin x + cos x at x = 0.

A: NOTE: Refresh your page if you can't see any equations. . take x=0 the point on the graph is

Q: 15. -- sin(2+ 2) dz

A: According to the given information, it is required to solve the give integral.

Q: Exercises to solve: 1. fx² In(x + 1) dx 2. (x2 + x – 2).cos4x. dx 3. Se-*sinxdx 4. S sinh2x.cosh5x…

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

Q: 2.1 Given that 1 sinh x show that dy cosh a. dx

A:

Q: Q3- Find: 1- Sv2 + sin3x . cos3x dx. 2-5* dx x5-2x sin 2x 2 sin x 4- f cscx cot x dx 4 cos x 5- S 3+…

A:

Q: 1. [cosh(x)] = sinh(x) dx

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: (b) Prove that : - 1) (coshx)2 - (sinhx)2 = 1 ,2) coshx - sinhx = e, 3) (cothx)2 -1= (cschx)2 %3D

A: Use hyperbolic  function

Q: x sin 2x sin cos x 2 X. Prove that 8 dx 2х - л

A:

Q: Compute cosh(x) and sech(x), assuming sinh(x) = 0.01. cosh(æ) = %3D sech(x)

A: