Q! O show th at [x sin'x + V1-x²] = sin'x sin x

Q: Compute cosh x and tanh x, assuming that sinh x = 0.8.

A: Here given as sinhx=0.8 Use the identity…

Q: Use the addition formulas to prove sinh(2x) = 2 cosh x sinh x cosh(2x) = cosh2 x + sinh2 x

A:

Q: Find the critical points of the function f(x)=sin2x−cosx on the interval (0,2π).

A: For critical points, first find the derivative of the function and equate it to zero.

Q: Integrate sin^2(7x)-cos(x)

A:

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

A: Given that  The following

Q: EX / Show that cosh x-sinhx=1

A:

Q: Evaluate S sinh x cosh x dx by introducing u-sinh x

A: Given that: ∫sinhx.coshx.dx

Q: tan zeecz coS z

A: Integration is the process of finding the anti-derivative function of the given function.…

Q: find the derivative.

A:

Q: integral of sin^5(3x)cos^3(3x)dx

A: The integral of ∫sin5(3x) cos3(3x) dx is sin5(3x) cos3(3x). If we need to find the integration then…

Q: 28. The derivative of In(x sin x) is 1. 1 + cotx а. b. C. d. tan x +x - +x sinx

A:

Q: Use the addition formulas to prove sinh(2x) = 2 cosh x sinh xcosh(2x) = cosh2 x + sinh2 x

A:

Q: Find the theoretical error bound in approximating SN" sin(x?) dx using Boole's Rule.

A: The theoretical error bound in approximating the integral∫0πsinx2dx using Boole's rule is shown…

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

A:

Q: x sin 2x sin cos x 2 Prove that: dx = 2x- T

A:

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

A:

Q: Find an equation of the tangent line to y sin 2x = x cos 2y at the point π/2 , π/4 . Do not…

A: First we will find Differentiation of given equation to get slope of the tangent line. Now equation…

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

A:

Q: (b) Prove that 1+ cosh x + cosh 2x coth x sinh x + sinh 2x 1+ cosh 2x + cosh 4x dx Hence evaluate…

A: 4b. Given,

Q: 9.) JE sin°(x)cos*(x)dx 11.) F sin°(x)cos²(x)dx 2

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

Q: 1. If f(x) = sinx, show that f(x + y)-f(x) = cos x siny - 2 sin x sin² 2 2Y-T

A: 1) Given f(x)=sin(x)

Q: 3. Show that sinh a + sinh y = 2 sinh cosh %3D

A:

Q: Compute integral sin(cos(x^2)) sin(x^2)x dx

A: Explanation is given below...

Q: 3. Prove that cosh (2x) = cosh²x + sinh²x %3D

A:

Q: Find the CRITICAL VALUES of f(x)=sin² x+cosx on the interval (0,2n)

A: Critical points are the points where the derivative of the function is zero or undefined. Our…

Q: x'+y° Q4. If u = sin- sin-'( -), Show that x+y ди X- ди + y- = 2tanx (2 + ду

A: If u=sin-1x3+y3x+y, then show that x∂u∂x+y∂u∂y=2tanx.

Q: Lim as x approaches 0  1-cos(5x)+sin(3x)/x

A:

Q: Find the fourth derivative of f (z) = 2" + In (x) – 160z. %3D Enclose arguments of functions in…

A:

Q: find integral sin 5 x  cos 2  x dx

A:

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

A: Given,           y=log1+tanx1-tanx

Q: Evaluate the integral. sin (12x) cos (7x)dx | sin (12x) cos (7x)dx =

A:

Q: Verify the identity cosh 2x = cosh2 x + sinh2 x

A: we have to verify the identity: cosh(2x)=cosh2x+sinh2x   L.H.S: cosh2x as we know that coshx=ex+e-x2…

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

A:

Q: Find the equation of line tangent of sin(2x) cos(3x)+ sec(x) TC At x = 6. %3D

A:

Q: 9. tan 4x sec24xdx d. cos 5xsin 5xdx e. x sin(3x² +1)dx f. [x* cos(2x' +1)dx

A:

Q: 57 ves y sinx and y=cosx from x=- to x=- 4 4

A:

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

A: Proved Coshx - Sinhx = e-x

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

A:

Q: Differentiate the function. f(x) = sin(x) In(3x) sin (x) f '(x) + cos (x) In (3x)

A:

Q: Find the linearization of ƒ(x) = sqrt(x + 1) + sin x at x = 0. How is it related to the individual…

A: To determine the linearization of the function at x=0.

Q: Find the first derivative of * In (sec y + tan y) - x - sinx = 2 O tany (cos.x + 1) O cos y (cos x +…

A:

Q: Find all points with x-values on (0, 2π), where tangent line to the graph of f(x) is horizontal.…

A: Find all points with x-values on (0, 2π), where tangent line to the graph of f(x) is horizontal.…

Q: (0)3-3x+2 sinh' (In x) cosh (In x) (E) sin(2x)dx dx N

A:

Q: limit x to 0 3x-3xcox/sin^2(3x)

A: Here given limit as Use the L'hospital rule which states

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 cosh 2 x - sinh 2  x =1

A: To prove  cosh2x-sinh2x=1

Q: 3 cos x Evaluate | dx by using substit ution method. (1+sin x)“

A:

Q: In (sinh x ), + tanh x j + 1-x k )dx 1- sinx

A:

Q: find tan x dx x² x² tan¬' x dx 19. 20. sin 3x cos 2x dx 22. sin 2x cos 3x dx S sin sin 8 sin 4t sin…

A:

Q: limit x approaches zero, sin^2x/x

A: Given,

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.