### Integrals substitution

Please see attachment for integrals substitution.

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Please see attachment for integrals substitution.

Which of the following conditions are necessary for a function f to be Riemann integrable on the closed interval [a,b], where a < b? I. f is bounded on [a,b]. II. f is continuous on [a,b]. III. f is differentiable on [a,b].

Please see the attached file for the fully formatted problems. 2. If f(x) =∫ (sin t)^(1/3) dt pi/2 --> x then at what value of x in the interval [0, 2pi] is f(x) a maximum?

Please show all of the steps needed to solve the 6 integrals and differential equations that are attached. A. 3/sqrt(6x - x^2) dx B. 4/(4x^2 + 4x + 65) dx Solve the differential equation. 1. dy/dx = (1 + e^x)^2 2. dr/dt = (1 + e^t)^J/e^t 4. (4 + tn^2x)y' = sec^2 x 5. y' = 1/(x*sqrt(4x^2 - 1))

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

Say the only tool you have available to you is a pocket calculator which performs addition, subtraction, multiplication, and division, accurate to 15 decimal places. Explain a practical way to compute: Integral from 0 to 1 of e^[-(x^2)] to within an error less than 10^-8. Prove that the method works.

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)

34.8 (a) Use integration by parts to evaluate 1 ∫ xּarctan x dx. 0 Hint: let u(x) = arctan x, so that u′(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].

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.

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

Please see the attached file for the fully formatted problem.

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

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

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

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 from x = 2 to x = 3 of: f(x) = 6x^2 - x - 6

Please see the attached equation. 0 ---> -2 ∫2x3 -4x dx

Please see the attached equation. 4 --> 1 ∫5x dx

Integrated from 0 to 4 : 3x^2 dx

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

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

Integrate 6 sqrt x dx.

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

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

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

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

Derive the composite midpoint method and composite error.

Please evaluate the attached by means of the residue theorem.

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

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