Recoll that co'e Find X = oree oSx. 1/2 *sin (eoi'x) de x sin

Q: sin(e)de 4+cos (e) In(4+cos'(e)) +c -sinh -sinh cos (0) sinh (cos (0)) +c 2/4+cos (0)

A: Use the substitution method and integrate the given expression.

Q: d) lime tan 0 sin 0 3. e) lim0 sin z-cos z+1 3z

A: We have to find the limits .

Q: Show that if z = x+ iy, then: | cosh-z|2 sinh x Hint: use the equation: | coshzf = sinhx+ cos"y

A: We have , z = x + iy We know that ,  coshz = ez + e-z2 Therefore , cosh (-z) = e-z + e-(-z)2 = ez +…

Q: Q.5/ Show that sin(z) = sin(x) cosh(y) + i cos(x) sinh(y)

A:

Q: (2 cos z-sin z)² dx Evaluate sin z

A:

Q: 3 sin° x -dx Evaluate ' cos° x

A:

Q: g(x) = 4ecoN " sin x, (0, ] COS

A: Given data:   The given function is: g(x)=4ecosxsinx The given interval is: [a,b]=[0,π].   The…

Q: cos z cot z = sin z

A: We have to prove cotz=coszsinz

Q: 3 Sin 3t-2 cos zt ; s()| こ what B the onti derivative Satifies re conditon is SLE) =>

A:

Q: dy - cosec'x. cos x (i) By differentiating 6 show that if y = cot x then sinx' dr (ii) Show that x…

A:

Q: s Sinx Cosy 1-Sin" x elx

A: To evaluate the given integral.∫sin x cos x 1-sin4 x dx

Q: Integrals Evaluate each integral.

A:

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

A:

Q: 2 + sin x Differentiate Y x + cos

A:

Q: (cos 0 -sin 0) d0 7. J sin 0 8. J (1 – tan 0)² de. 9. sec2 0 tan² 0 d0.

A: 7. Given integral, ∫cos θ-sin θsin θdθ

Q: 1+Sinx-Cos %3D 1+Sinx+ COSK 1+ Sinx+Cessx %3D 1+Sinx-CosX Ni = 2.5 Newtons methaw

A: Newton raphson method

Q: Using an appropriate substitution, cos x-sin x the integral 3. dx equals: 0 cosx+sin x O none of…

A:

Q: B) If sinh e V3 Find cosh 0, tanh 0, coth 0.

A: The function sinh(θ) is called the hyperbolic sine function. It is defined as sinh(θ)=eθ-e-θ2.…

Q: ) Sea z = sen(x – at) + cos(x + at). Verifique si a2. ||

A:

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: 3. KIN 2 13² cos x V1+sin x dx

A: The given integral, ∫0π2cos x1+sinxdx

Q: cos x 4. Compute the integral 1+(sin x)? Hint: u-substitution

A: the given expression is: ∫0π2cosx1+sinx2dx let the given integral be I. therefore,…

Q: c Functions. Prove the identity cosh x – sinh x = e x-

A: We have to show the given condition

Q: en(cos (Tix)e) + हाक %3D COS

A: Given that y=ππx ln(cos(πx)e)+sin3π5,   we have to find implicit derivative of y.

Q: 3 + cos 3x dx Evaluate sin 3x

A:

Q: Sin'z cos' de

A: Given: ∫sin12x cos5x dx  ……1

Q: 1 + sin x Differentiate y = x + cos x

A: As we know that: The Quotient rule: fg'=f'·g+g'·fg2 The Power rule: ddxxn=nxn-1 Given:…

Q: Prove the pollowing identity.. (sin x t cos L cas x)- sin 2X-CoS T Sin'x + Cosx

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Q: e sin e Cos e tan 0 a 0.5446 0.2924 0.4452

A:

Q: H(x) = sin(esinh~1 *) %3D

A:

Q: e* +e* cosh(x) = cosh(0) = ? 33- e* -e sinh(x) = → sinh(0) = ? 2

A:

Q: cos ax dx = xe", a 2 0. (or) Evaluate x' +1 cos ax 2: Prove that x +1

A:

Q: Recoll that e ošx Find S orecosx. 1/2 x sin(coi'x ) de

A: ∫012xsincos-1xdx

Q: y = (2x³ + tan(x)) dy Find da dy da Type sin(x) for sin(x), cos(x) for cos(x), and so on. Use x^2 to…

A:

Q: x2 cos x 6. Find for y = ln: dx sec x а. 2x – sin x – tan x sec x b. + cot x - tan x 2 cot x – tan x…

A: We need to differentiate the equation with respect to x.

Q: et +e» et Рe-* cosh(x) sinh(x) 2

A: Given that:- coshx=ex+e-x2 sinh(x)=ex-e-x2

Q: Evaluate sin z -dx /4 - coS I

A: Use the substitution method and first simplify the integration than integrate.

Q: lim E(cos (27x;))Ar over [0,5] COS i=1

A: The definite integral is defined as  ∫abf(x)dx=limn→∞∑i=1nfxi*ΔxWhere, Δx=b-an, xi*=a+iΔx The…

Q: 1+ sin 0 cot e Prove : 1- sin 0 (cos ece – 1}

A: Given:                            1+sin θ1-sin θ=cot2θcosec θ-12 Solve left hand side:…

Q: 1IF Sine2-0.9, Find the valu e of Sino + Sin(e+2 + Sin(e+47

A: Our guidelines for multiple questions,answer only first question

Q: (4) Use diffential equations to prove the trig identity sin(r + a) = sin(r) cos(a) = sin(a) cos(x)…

A: Given: sin(x+a)+sin(x)cos(a)=sin(a)cos(x)

Q: 1 dx 1+sin x+cosx Evaluate:

A: Given integral, ∫11+sinx+cosxdx

Q: S sec²0 do 1+tano seco tane de seco - 1 The integral O 1: S S x dx x²+√√√x²-1-1 coso do 1-sino sino…

A:

Q: 8(cos² z – 1) sin z cos z – sin z

A: Given,           8cos2x-1sinxcosx-sinx

Q: evaluate using the appropriate method or combination of methods.

A:

Q: By using Cauc hy laws find= Cos X.dx IT (x?+)) Sinx cos A) dx COS X e)

A:

Q: y = arctan dy Calculate da Type sin(x) for sin(x), cos(x) for cos (2), and so on. Use x^2 for a",…

A:

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

A: To show that sinhx+y=sinhx coshy+coshx sinhy

Q: Question attached as photo

A: To verify the values of the given definite integrals

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.