Prove that for all x € [1, 00), – sinh (- |cosh'(x)|) = V -1. %3D Ouest!

Q: dy +y = zy dr

A: Given equation is: xdydx+y=xy3

Q: Determine if the improper integral converges or diverges: 1 dx. 1+x²

A: Our Aim is to determine whether the improper integral given below converges or diverges∫0∞11+x2 -(i)

Q: 4 18. Evaluate the indefinite integral / dx. 3 X,

A: Given integral to find ∫4x3dx To integrate this first take the constant out 4∫1x3dx Applying the…

Q: Conclude on the convergence or divergence of sin x dx cos x

A: Since function is not continous at x=pi/2. Hence we break the limit of integral at x=pi/2. This…

Q: Suppose that o(x, y) = e"-y and ø(ln 2, In 2) linear approximation to estimate o(ln 2 + 0.1, lIn 2 +…

A: Given that Φ(x,y) = ex-y and Φ(ln2,ln2)=1. Now by using the technique of linear approximation we…

Q: 2. Evaluate the integral / sin'at dt

A:

Q: 3. (a) Compute ,12x – 1| dx in two ways. i. By using properties of the definite integral and the…

A:

Q: Find where ln((1+z)/(1-z)) is not analytic.

A: To find: The region where the function ln(1+z)(1-z)), where z∈ℂ is not Analytic.

Q: Find out if the improper integral converges or diverges. If the given converges, find its value.…

A: Solution

Q: Prove that sinh−1 t = ln(t + √t2 + 1 ).

A:

Q: d Prove In Jax| dx In x for any constant a. dx

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Q: to tal differential of eind the the り23 function ] 2=e + tan Z= Sin 2.

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Q: O Use a linear approxi mation (differential) to estimate 3999

A:

Q: Show that integral 0 to infinity e -x 2 is convergent.

A:

Q: - Evaluate the integral dx using Simpson's th rule e* – 1 3 -

A:

Q: 7. Evaluate the improper integral. -kx e (k: constant)

A: Please refer to the image below.

Q: Determine whether the improper integral diverges or converges. dx 1.co (2x-1)* 00 dx 2. x +1

A: To check weather the given Integral is converges or diverges.

Q: Use differentials to estimate sin(0.1). A) 0.0017 0.0099998 0.0998 D. 0.1

A: Give: To using differential s to estimate value of Sin(0.1)

Q: dy dr

A:

Q: Evaluate the integral if it converges. Enter the exact answer, or enter na if it diverges. +00 -dx =…

A: Let I is the value of given Integral.

Q: Prove that coshz =log ztv= {Hint : cosh z =e} 2.

A: Use the hint and prove this.

Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: Show that for every 0 < p < 1 the integral 1 1 sin – dx xP is both improper and convergent.

A: Answer

Q: Use the Beta and gamma function -1 n! 1- Prove that x™(ln x)"dx (m+1)n+1

A:

Q: Consider f(x) = sin(2x) + cos(4x), 0 ≤ x ≤ 1. .......Integrate f over the given interval of x, using…

A: Solution:

Q: Evaluate the definite integral. ∫(lower bound=π/4)(upper bound=0)sin(4t)dt

A: Given,             ∫π40sin4tdt

Q: What is the integral of sin^5 x cos^3 x dx if the lower limit is zero and the upper limit is T/2?

A:

Q: Consider f(x) = sin(2x) + cos(4x), 0 ≤ x ≤ 1. Integrate f over the given interval of x, using…

A: Consider the provided question, fx=sin2x+cos4x, 0≤x≤1. We need to Integrate 'f' over the given…

Q: (ii) Hence, by using Leibniz Rule for Differentiation Under the Integral Sign, compute the value of…

A: Leibniz Rule for Differentiation under integral sign:- ∂∂t∫atbtfx, tdx=∫atbt∂f∂tdx   + fbt, t∂bt∂t-…

Q: Use an integrating factor to solve the IVP y'+ytan x = cos x, with IC (x, , Yo) = (0,7), %3D and…

A:

Q: integral of (e^2x)(sin3x) using u=e^2x

A: Given, ∫e2xsin3xdx    We have to find the integral

Q: 6. The improper integral 1 dx converges to х -5 попe 2 1 4

A: We have to find

Q: 7 In x (1) Evaluate the definite integral dx.

A:

Q: Show that for every 0 < p < 1 the integral 1 1 sin = dx xP is both improper and convergent.

A:

Q: 3.12 Еxpress lim n00 n k3D cos 1+ as an integral. (Be sure to include the lower and upper limits|of…

A: Given: limn→∞ 2n∑k=1ncos1+2kn To convert this into the integral , use the following method

Q: calculate the limit of ln(x) - (e)^x  x goes to + infinite

A:

Q: I Braluate th FOllowing integralis 3+cos (logqx) dx x sin (loga x )

A:

Q: Evaluate the indefinite integral as an infinite series. | arctan(x2) dx Σ + C n = 0

A: Answer

Q: Show that integral 0 to infinity  x2 e -x2  dx = 1/2 integral 0 to infinity e -x2  dx

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Q: 5. Express the integral: e- dr in terms of gamma function.

A:

Q: ons z = sin y arn

A:

Q: y²

A:

Q: b) Use it in oonjunction with Simpson's 1/3 nule to integrate the same function for ive egmenta…

A: Introduction: Simpson's rule is a method to approximate a definite integral. The interval must be…

Q: a. Find the root mean square value of tanh x, for 0 < x< 2.

A: we know the formula of root mean square, rms = [ (1/b-a) (integration of |f(x)|2 from a to b) ](1/2)…

Q: 2. Prove that g(x) = sinh x + x – 1 has a zero in [0, ln 2].

A: To prove that gx=sinh x+x-1 has a zero in 0, ln 2.

Q: Using Simpson's (1/3) rule with h = 1, evaluate the integral: 7 = f x² logx dx 3

A:

Q: Only one of these calculations is correct to evaluate the definite integral Vx+1 dx Using the…

A:

Q: Use the Romberg integration to evaluate. 3 xe*dr With an exactitude e =0.5%

A: Given ∫03xexdx whose step error should be εs=0.5%. We use the Trapezoidal rule…

Q: prove using the IVT. e^x + ln x = 0 has a solution.

A:

Q: arc sin z = -i log(iz + V1- 22).

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.