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Definite and indefinite integrals

Thank you in advance for your help. Evaluate the following definite and indefinite integrals: (1)The integral of (2x)/([x^2]+1)dx (2)The integral of [(arctan(x))/([x^2]+1)]dx (3)The integral of sqrt([x^3]+[1x^5])dx (4)The integral of (2+x)/([x^2]+1)dx (5)The integral of (2x)/([x^4]+1)dx (6)The integral from 0 to 2 of (x

Integrals : Rate of Change Word Problem

Water flows from the bottom of a storage tank at a rate of r(t)=200-4t liters per minute, where 0 is less than or equal to t and t is less than or equal to 50. Find the amount of water that flows from the tank during the first 10 minutes.

Integral of a Contour

Calculate the following integrals: ∫ from 0 to ∞ x^¼/(x²+9) dx Please see attached for proper format.

Residues and Closed Contours : Solve the Integral

Calculate the following integral... Please see attached for full question. Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C. Let . By residual Theorem

Integrals : Riemann Sum with Diagrams

This question has me going around in circles. I can't make the Sigma symbol on the computer, so I used the word "Sigma" instead. For (c), n is above the Sigma symbol and i=1 is below it. (a)Find an approximation to the integral as 0 goes to 4 of (x^2-3x)dx using a Riemann sum with right endpoints and n=8. (b)Draw a diagram

Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Consider approximating integrals of the form... in which f(x) has several continuous derivatives on [0, 1] a. Find a formula... which is exact if f(x) is any linear polynomial. b. To find a formula... which is exact for all polynomial of degree ≤ 3, set up a system of four e

Lowest Common Multiples and Diophantine Equations

Please solve the following problems: 1. Compute the following ... 2. Let Fm be the set of all integral multiples of the integer m. Prove that ... 3. Draw the graphs of the straight lines defined by the following Diophantine equations ... 4. Prove that every integer is uniquely representable as the product of a non-negati

Multiple Integration, Area, Center of Mass, Centroid and Jacobian

1.Given the region R bounded by y=2x+2 , 2y=x and 4. a) Set up a double integral for finding the area of R. b) Set up a double integral to find the volume of the solid above R but below the surface f(x,y) 2+4x. c) Setup a triple integral to find the volume of the solid above R but below the surface f(x,y)=-x^2 +4x. d) Set

Integral of a Principal Branch

Show that the integration from -1 to 1 z^i dz = ((1+e^-pi)/2)*(1-i)where zi denotes the principal branch... (See attachment for full question)

Euclid's Division Lemma and Fundamental Theorem of Arithmetic

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j ? qk is positive. 2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle: Every non-empty set of positive integers has a least element.

Volume of a Hypersphere

Finding formulas for the volume enclosed by a hypersphere in n-dimensional space. c) Use a quadruple integral to find the hypervolume enclosed by the hypersphere x^2 + y^2 + z^2 + w^2 = r^2 in R^4. (Use only trigonometric substitution and the reduction formulas for ∫sin^n(x)*dx or ∫cos^n(x)*dx.)

U-Substitution, Integration by Parts and Differential Equations (8 Problems)

Please show all of the steps needed to solve the 8 integrals and differential equations that are attached. The integral of x(cos(x) dx The integral of (x^3) sin(x) dx The integral of t(csc(t))cot(t) dt The integral of arctan x dx The integral of e^2x sin(X) dx Solve the differential equation. y' = xe^x2 dy/dt = y

One Dimensional Riemann-Integrable

Q. Show that f is Riemann-integrable. What is ∫[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.

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

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

Changing variables in multiple integrals

Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2. Choose the order of integration which make the limits simplest THIS MESSAGE IS ADDRESSED TO ANY TA: I found something , I just want you

Function and Differential Equations

See attached explanation Differential equations are not my strong suit now. Please explain in a simple way each step from the integral 1/F dF to the final answer. Show and tell how you get from step to step. On problem 35 please answer and explain this in the simplest way you can for me to understand please. Step by

Limits of iterated integrals (parallel axis theorem)

Prim is primitive! In genral the moment of inertia around an axis( a line) L is: Isubl=double prim (dist(.,L)^2*delta*dA) The collection of lines parallel to the y axis have the form x=a .Let I=Isub(y) be the usual moment of inertia around the y axis I= double prim of x^2*delta*dA Let I(bar) be the moment of ine

Integration: The Limit of a Sum

Find by the method of summation the value of : a) The integral (from 0 to 1) of the square root of x. (dx) b) The integral (from 1 to 4) of 1 divided by the square root of x. (dx) Please view the attachment for proper formatting.

Calculus Functions to Evaluate an Integral

Please help with various Calculus questions. You do not need to show your work for this one because I would simply like to compare your answers with mine so that I am sure that I did everything correct on mine. Please just write your exact answer after each number. I will know which problems I will have to study in detail w

Partial Derivative and Double Integral

The problems are attached 1 -5 based on Chapter Partial Derivative - (Maximum & Minimum Values and Lagrange Multipliers 1. Locate all relative maxima, relative minima, and saddle points of the surface defined by the following function. 2. Consider the minimization of subject to the constraint of (a) Draw the