If y= tanu, u=v-- -, and v=ln x, dy at x= e? dx what is the value of

Q: Find y′ if y = tanh−1(sinh x2).

A:

Q: xe 2, x Cosz dx

A: Integration by parts, ∫udv=uv-∫vdu u is the first function Given integral, ∫x exdx

Q: Find the indefinite integral. sinh(6 – 7x) dx

A: Consider the integral  ∫sinh(6-7x)dx

Q: S" S"(sinx + cosy) dx dy

A: our objective is to evaluate the integration

Q: 8. if y =; sinh(2x + 1) 9. ſ cot* 0 de. 10. * if y = cot-()- tan-1x.

A:

Q: If x = tan - log y a d'y dy Show that (1+x²)- +(2x- a)-0 dx dx?

A:

Q: Which of the following is correct? " 17 3 (3 sinh-(x)) = (coth"(x) = (지 < 1) V - 1 dx Option 1…

A:

Q: 1. Ye = cost- sec(at) %3D ans: =-2sin t - asec(at) tan(at)

A:

Q: Find the indefinite integral. ∫ sinh x/(1 + sinh2 x) dx

A:

Q: L(cos(at)sinh(at))

A:

Q: sinh In(2+2x) dx? x+1

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: By using the exponential definitions of sinh and cosh. show that cosh (t) - sinh (t)

A:

Q: dx х V1-4 ln?х

A:

Q: If y- sinhx then prove that = dx

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

Q: a. y = cot¯(Ve") tanh(e") -1

A: Since you have asked multiple questions in single request so we will be answering only first…

Q: Find the indefinite integral. sinh(4 - 7x) dx

A: evaluate the indefinite integral ∫sinh4-7xdx

Q: TT sin 5. Approximate dx by Simpson's rule with seventh ordinates?

A: Simpson's Rule:    ∫abf(x)dx=∆x3f(x0)+4f(x1)+2f(x2)+...+2f(xn-2)+4(fn-1)+f(xn) Where, ∆x=b-an We…

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

A:

Q: Find if y = e* In(sinh x) dx

A:

Q: S sin Vat dt

A: Let the given integral be: I=∫sinat dt

Q: 42. y=+on [1/2,2] 6

A:

Q: if Pen) = Sinh(nw) find F(Z)=?

A: There is an exponential representation of sinhx. Whenever we analyse various properties of sinhx, we…

Q: 5. If cosh 2x then the positive value of 4 sinh 2r is

A: Here we will use the relationship betweencosh2x and sinh2x to find the value of sinh2x from the…

Q: find dy / dx for Hhe tellowing in (sec EX) In (cosp?}

A:

Q: What is the definite integral of sin^5 (x) cos^3 (x) dx if the lower limit is zero & the upper limit…

A:

Q: If y- sinhx then prove that dy dx

A:

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: Find the indefinite integral ∫sinh(1 − 2x) dx

A:

Q: eteR MiNe SiN IN°()COS)dx

A:

Q: Find , if y = x sinh x - cosh x %3D dx

A:

Q: Find the indefinite integral. sinh(4- 5x) dx

A: To find:∫sinh4-5xdx

Q: xcos( x), sin(x) 8. dx = LHIN

A:

Q: Find the indefinite integral. S sinh (1 – 2x) dx

A:

Q: Show that coshr+ sinh a = e and cosh r- sinhr = e. Hence, deduce UTMOU cosh r- sinh z = 1.

A:

Q: 1. Use the fact that coshx = (e* + e) and sinhx = 1 NIH (ex - e-*) to solve for 1 coshx+ sinhx dx

A: ∫exdx=ex+c

Q: If y = (sin-' x)², show that (1 – 2*) dy dr? dy 2. dx

A:

Q: Determine the length of the arc y=ln(cos x) for 0≤x≤π3.

A:

Q: Find the indefinite integral. ∫sec x/2 dx

A: Given ∫sec x/2 dx. We need to evaluate the above integral.

Q: S tanʼx dx

A: Answer is given in next step.

Q: The approximate value of dx by using Simpson's rule is 0.3480, then the error in the approximation…

A: Given Approximate value of the integral by Simpson's rule(Vs) = 0.3480 Now we will find the actual…

Q: Find the indefinite integral. ∫4 arccos x dx

A: To evaluate the integral: ∫4cos-1x dx Integration by parts states that: ddxuv=v·ddxu-∫ddxv∫udxdx…

Q: calculate the integral.

A: Given integral is We will use the following substitution to evaluate the integral

Q: tanh 2x In (cosh 2x)dx

A:

Q: if 2 tanh x.In x = sinh' (sin x)+logcos dx dy Q5/ Find

A: To differentiate the given function.

Q: 10.y = Arc cot (sinh3x) 11. y = cosh?4x - cos 12. y = In (sinh4x) cosh8x +x² 2

A: To find dy/dx: 10.y = arc cot ( sinh3x ) dy/dx = [1/(1+sinh23x) ]( cosh3x) 3  = (3 cosh3x) /…

Q: Find the deriva tive Of 6- sin (xy)+ e'

A: Note:- It is not mentioned in question for derivative find with respectvto what variable(x or y) so…

Q: dy/dx at the same point (1,0) is

A: Given: The equation is y = ln ( x2 + y2 ) and the point (1 , 0).

Q: Cualuate the definle integial. x33 COs (xH) dx

A:

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.