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# Integrals

### Indefinite Integral Square Root Functions

Find the indefinite integral (3-x)/sq root of 9-x^2 dx(dx would be in the numerator). I tried to split this problem apart. First part was: The integral of 3/sq root of 9-x^2 dx and found 3 arcsin x/3 + C, then Second part was: The integral of -x/sq root of 9-x^2 dx and found -3/4 -x + C. I then put them back together to ge

### Analysis Proof Absolute Value

Note: If you have already answered this exact question please do not answer it again. I would like an answer from a different T.A. Thanks Say abs = absolute value. Suppose that the function f:[a,b]->R is Lipschitz; that is , there is a number c such that: abs(f(u) - f(v)) <= (c)abs(u-v) for all u and v in [a,b]. Let P

### Real Analysis : Proof using an Integral and Mean Value Theorem

For numbers a1,....,an, define p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that: (a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0 Prove that there is some point x in the interval (0,1) such that p(x) = 0

### Picard's Method of Successive Approximations

Please see the attached file for the fully formatted problems. Attached is a file with a three part successive approximation problem. The following problems are to use the method of successive approximations (Picard's) [EQUATION] y x y fty tdt =+&#8747;n&#8722; with a choice of initial approximation other than y0(x)=y0

### Infinitely Differentiable Function that is Not Analytic

Use the given information: the functions g:[a,b]->R and h:[a,b]->R are continuous with h(x) >= 0 for all x in [a,b], and there is a point c in (a,b) such that: the integral from a to b of h(x)g(x)dx = g(c) times the integral from a to b of h(x)dx. to show that the Cauchy Integral Remainder Theorem implies the Lagrang

### Green's Theorem Evaluated Integral

Apply Green's Theorem to evaluate the integral over C of 2(x^2+y^2)dx + (x+y)^2 dy, where C is the boundary of the triangle with vertices (1,1), (2,2) and (1,3) oriented in the counterclockwise direction. Also check the result by direct integration. Please show detailed working so I can follow the steps of the working.

### Indefinite Integrals for Anti-Derivative

Find the indefinite integrals (anti-derivatives): Find the indefinite integrals (anti-derivatives): a.) x / (x +2) dx I found ½ ln + + C as an answer - is this correct? b.) 1 / (x +2) dx I found 1/x arctan /x + C as an answer - is this correct? (I said that a = x, u = , du = dx )

### Real Anaylsis

Let f: [a,b] be mapped onto the Reals be a function that is integrable over [a,b] and let g: [a,b] be mapped onto the Reals be a function that agrees with f except at finitely many points. Is g integrable over [a,b]? Why or why not?

### Function integrable proof

Let f: [a,b] mapped onto Reals be a nonnegative function that is integrable over [a,b]. Then the integral from a to b of f = 0 if and only if greatest lower bound of f (I) = 0 for each open interval I in [a,b].

### Sketching and Double Integration

Please see the attached file for the fully formatted problems. I.A. Sketch the following region in the x-y plane: R: 0<x<b^2 : x^1/2 < y< b B. Set up integral R for (e^-y2)/y dA in two ways.

### Set up triple integral for volume of cone.

Please see the attached file for the fully formatted problems. Set up triple integral for volume of cone, do not evaluate.

### Sketch the curve and set up double integral for bounded area.

Please see the attached file for the fully formatted problems. Sketch the curve r = 5 - 3cos(theta) and set up double integral for bounded area in the third quadrant.

### Evaluating an indefinite integral

Evaluate the following indefinite integral. int[(x^a)sqrt(r+tx^(a+1))]dx, (t not=0, a not=-1). (See attachment)

### Evaluate an improper integral using Jordan's Lemma

Use residues to evaluate this improper integral Int(from 0 to inf)[cos(ax)/(x^2+1)]dx (a>0) (See attachment for better description.)

### Triple Integrals : Finding Volume of Solids with Boundaries

1) Evaluate the triple integral e^(1-(x^2)-(y^2)) dxdydz with T the solid enclosed by z=0 and z= 4-(x^2)-(y^2) 2) Find the volume of the solid bounded above and below by the cone (z^2) = (x^2) + (y^2), and the side by y=0 and y= square root(4-(x^2)-(z^2))

### Integration: Cauchy-Schwarz Inequality

Suppose that the functions g:[a,b]-> R are continuous. Prove that: The integral from a to b of gf <= (the square root of the integral from a to b of g^2) multiplied by (the square root from a to b of f^2)

### Double Integral : Integrating with respect to two variables.

Please see the attached file for the fully formatted problem.

### Convergence or Divergence, Absolute or Conditional of Summation

Test for convergence or divergence, absolute or conditional of a summation. Sigma infinite over and n=1 under times 2n+1/n^2+2.

### Quick Calculation of Laplace Integral

Please see the attached file for the fully formatted problem. Construct the quickest method to calculate the Laplace Integral. I = S e^(-x^2) dx infinity --> infinity

### Integration: Standard Partition, Integrable over a Range

Questions on integration, see attachment.

### Integration: Standard Partition and Integrable over a Range

Please see the attached file for the fully formatted problems.

### Integration: Stirling's Formula

Questions on integration. Please see the attached file for the fully formatted problems.

### Integral test: Convergence or divergence of a series.

Use the integral test to determine the convergence or divergence of the series: En=1 2 / (3n + 5)

### Integration of Polynomial Fractions

Integral x-1, divided by x to the 3rd + x squared to dx. x-1 ----- X to the 3rd + X to the 2nd ( all dx)

### Integration and Leibniz's Formula

For problem #1, its the integral from o to infinity (the symbol for infinity for that problem was cut off)

### Evaluating Integral Functions

I'm taking a DE calculus class and I'm having problems figuring out the logic in solving some of the problems. The given integral is improper because both the interval of integration is unbounded and the integrand is unbounded near zero. Investigate its convergence by expressing it a sum of two intergrands-one from 0 to 1 an

### Double Integration Transformation

Evaluate the double integral Transform the double integral of (i) using plane polar coordinates Show that the 3 x 3 determinant See attached file:

### Integrals: Cost function and Marginal Cost

Given ist the following cost function: k(x)=x^3-9x^2+29x+35 x= quantity k= cost Question 1: At what quantity is the minimum of the marginal cost? Question 2: What is the increase of cost if the production is increased from 3 to 4 (integral)?

### Evaluate integrals

Please see the attached file for the fully formatted problems. Evaluate the following integrals. S (4x^3 -2x - (2/x^3) dx S (1/2x^1/2) dx 1-->0 S ln x dx

### Calculating integrals: Sines and Cosines

Integrate {cos(x)cos(3x)cos(5x),x}