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# Integrals

### Indefinite integrals found

1. Find the indefinite integrals 2. Suppose that the rate of increase of paper production in the US for recent year is given approximately by... (see attachment)

### Integral

Please evaluate the integral attached

### Integration by parts

34.8 (a) Use integration by parts to evaluate 1 &#8747; x&#1468;arctan x dx. 0 Hint: let u(x) = arctan x, so that u&#8242;(x) = 1/(1+x2). (b) If you used v(x) = x2/2 in part (a), do the computation again with v(x) = (x2+1)/2. This interesting example is taken from J. L. Borman[6].

### Integral

Evaluate the integral from 0 to inf. of [cos(3x)-cos(5x)]/x^2 dx

### Definite integral

Evaluate the integral from 0 to infinity of X^2/(X^4+2X^2+1) dx

### One Dimensional Riemann-Integrable

Q. Show that f is Riemann-integrable. What is &#8747;[0,1] f(x)dx? (Hint: What's the set of discontinuity of f? Does it have Vol1-zero?) Please see attached for full question.

### Integral, Continuity and Limits

Please see the attached file for the fully formatted problems. Q: Suppose and are continuous and F(x) = Let (a) prove that f'(x) + g'(x) =0 for all x (b) Prove that f(x) + g(x) = /4 for all x. Deduce that

### Evaluate the Integral

Please see the attached file for the fully formatted problem.

### Integration : Fubini's Theorem

Use attached to solve the following question by integrating over an appropriate rectangle. Assume f is class C2 Prove the following theorem by Fubini's Theorem. Please see attachment. For f of class C2 Left Hand side: Right Hand side: Use above to solve the following question by integrating over an appropri

### Potential Equation on a Disc : Dirichlet

Solve the potential equation on a disc... (See attachment for full question)

### Integration : Class C2

Calculate to show, for f of class C2 ... {see attachment} What is the integral on the right equal to {see attachment}

### Fubini Type I : Interpret Iterated Integrals as Triple Integral

Interpret the attached iterated integrals as a triple integral for the appropriate region {see attachment}, sketch {see attachment} and change the order of integration so that the innermost integral is taken with respect to y. (f is continuous) ... **See attachment for complete question.

### Evaluate the Definite Integral

Evaluate the definite integral from x = 2 to x = 3 of: f(x) = 6x^2 - x - 6. See the attached file.

### Definite Integral Equations

Please see the attached equation. 0 ---> -2 &#8747;2x3 -4x dx

### Integrate using chain rule with inserting coefficient.

Integrate (3x+1)^1/2 dx rewrite as Integrate (3x+1)^1/2 1*dx use chain rule with inserting coefficient.

### Integrate using the chain rule.

Integrate (x^3+2x^2)^8(3x^2+4x)dx. Use the chain rule.

### Integrate the Function

Integrate 6 sqrt x dx.

### Integrate 12/x^4dx

Integrate 12/x^4dx

### Integrate (6x^2-4x^3+5x^4)dx

Integrate (6x^2-4x^3+5x^4)dx.

### Find the area under the curve.

Find the area under the curve y=-x^2+3x from x=1 to x=4

### Continuous Functions, Fundamental Set of Solutions

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I. (a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation. (b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu

### Solve the ODE

Use words to describe the solution process. Typeset solutions. Work is to be done without the aid of a calculator or computer. Show all steps. For example, if you integrate by parts, show all the steps of integration. If you use an integral table, state that as well. Find the solution to y''' + 2y'' + y' = 0 satisfying

### velocity expression satisfies the partial differential equation

Thank you in advance for your help in solving this problem. See attached problem statement.

### Calculating Traction from Stress Tensor Matrix

NOTE: in part A, the traction is just the integral of the dot product of T and n. 7 0 -2 The stress at point P = 0 5 0 -2 0 4 I want to know the traction vector on the plane at point P with the unit normal n = (2i1, -2i2, 1i3)/3

### Numerical Integration: Composite Midpoint Method and Error

Derive the composite midpoint method and composite error.

### Residue Theorem and L'Hopital's Rule

Please evaluate the attached by means of the residue theorem.

### How to Find the Integral of a Function

See attached. Let g(a) be the real solution to x+x^5=a

### Joint Density

The joint denisty function of X and Y is given by [see attached] (a) Find E(X) (b) Find E(Y) (c) Show that Cov(X,Y)=1 *(Please see attachment for complete problem)

### Area Between a Curve

Please assist me with the attached problems relating to finding the region within a curve. 3. (a) Obtain an expretsian far Calculating the area between the curve y=2?x+x2 and the u-axis far 0 <x< 2 by dividing the area up into 2n strips of equal width (each strip will have width 1/n) and then taking the limit as n ---> infini

### Centroid-triple integrals

Find the centroid of the first octant region that is interior to the to the two cylinders x^2+z^2=1 and Y^2+Z^2=1 centroid for x y and z are x'=1/M*triple integral of x^2*dV y'=1/M*triple integral of y^2*dV z'=1/M*triple integral of z^2*dV