### Integration by Parts

(Integrate from 0 to 1) [(x^2+1)e^(-x)]dx Please show steps.

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(Integrate from 0 to 1) [(x^2+1)e^(-x)]dx Please show steps.

A 10-ft trough filled with water has a semicircular cross section of diameter 4 ft. How much work is done in pumping all the water over the edge of the trough? Assume that the water weighs 62.5lb/ft^3.

1) Evaluate. ∫ sinh^6 x cosh x dx 2) Evaluate. ∫ [ (√x+4)^3 / 3√x ] dx 3) Evaluate. ∫ x^2 sin(2x) dx 4) Evaluate. ∫ sin^5 x dx Please see attachment for actual sample problems.

3. Solve the wave equation, ∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞ With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1) 4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coeffici

1. Evaluate: ∫2cos2 xdx 2. Figure 12.1 y = 9-x2 , y=5-3x Sketch the region bounded by the graphs of Figure 12.1, and then find its area. 3. Figure 13.1 1?0x4dx Approximate the integral (Figure 13.1); n=6, by: a) first applying Simpsonfs Rule and b) then applying the trapezoidal rule. 4. Find

LetR be the region bounded by the graph of f(x)=3x^2+6x and g(x)=18x-5x^2. 1) determine the area of R 2) Determine the volume of the solid of revolution formed when R is revolved about the line y=18. Please answer in detail.

Evaluate the integral. (integral from 0 to 1) dt/sqrt[16t^2+1] Use Hyperbolic inverse and show steps please!

Evaluate the integral. (integral sign) e^(2x)dx/sqrt[1-e^(4x)] Please show steps.

Evaluate the integral. (integrate from 0 to 1) dt/sqrt[16t^(2)+1] Please show steps

Please see the attached file for the fully formatted problems.

1)Figure 11.1: 0<=x<=(pie)/2 R is bounded below by the x-axis and above by the curve y = 2cos(x), Figure 11.1. Find the volume of the solid generated by revolving R around the y-axis by the method of cylindrical shells. 2)Figure 15.1: y= 1/(x^2+4x+5) R is the region that lies between the curve (Figure 15.1) and t

See attachment and show work. 1. Find a function f(x) = x^k and a function g such that f(g(x)) = h(x) = (3x + x^2)^0.5; 2. Express the distance between the point (3, 0) and point P(x, y) of the parabola y = x^2 as a function of x 3. Determine whether converges or diverges. If it converges, evaluate the integral

Differentiate the function f(x) = ln(2x + 3). Find . lim e^ 2 x/(x+5)^3 →∞ Apply l'Hopital's rule as many times as necessary, verifying your results after each application. Evaluate ∫ x sinh(x)dx . Determine whether 2 ∫ (x / ^(4-x^2)) (dx)

Use implicit differentiation to find an equation of the line tangent to the curve x3 + 2xy + y3 = 13 at the point (1, 2). What is the maximum possible area of a rectangle inscribed in the ellipse x2 + 4y2 = 4 with the sides of the rectangle parallel to the coordinate axes? 3 . Evaluate ∫ dt / (t+1)^

Evaluate ∫3x+3 / x^3-1 (dx) Use trigonometric substitution to evaluate ∫1 / ^/¯1+x2(dx) Determine whether converges or diverges. If it converges, evaluate the integral. ∞∫-∞ 1 / 1+x2 (dx)

Evaluate ∫(^/¯x+4)^3 / 3^/¯x(dx) ∫x2sin2x dx ∫ sin5xdx

Evaluate: ∫ sinh6 xcosh xd x Given ?(x)= csch^-1 1 /x2 find ?'(x) Given ?(x)=log10x find ?'(x).

Find an upper and lower bound for the integral using the comparison properties of integrals. 1∫0 1 /x+2(dx) Apply the Fundamental Theorem of Calculus to find the derivative of: h(x)= x∫2 ^/¯u-1dx Evaluate: 4∫1 (4+^/¯x)^2 / 2^/¯x (dx) Evaluate: ∫2cos^2 xdx Sketch

Please see the attached file for the fully formatted problemsUse Green's Theorem to evaluate the integral (6y + x)dx + (y + 2x)dy C: The circle (x-2)^2 + (y-3)^2 = 4.

Show that if R is a region in the plane bounded by a piecewise smooth simple closed curve C then area is given by.... Please see the attached file for the fully formatted problem.

Evaluate a. b. c. d. An object is moving so that its velocity after t minutes is meters per minute. How far does the object travel from the end of minute 2 to the end of minute 3? a. 31 meters b. 61 meters c. -29 meters d. -9 meters

See attached word doc.

Given a spherical Bessel function J1(w) = (1/w^2)*(Sin(w)-w*Cos(w)) Show that (pi)*x/2, -1<x<1 integral of J1(w)*Sin(wx)dw from 0 to infinity = 0, |x|>1 Need a through explanation!!

1. Use the composite Trapezoidal Rule with indicated values of n=4 to approximate the following integrals See Attached file for integrals. 2. Use the Excel programs for Simpson's composite rule to evaluate integrals in Problem 1. 3. Use Gaussian Quadratures with n = 2, n = 4, n = 5 to evaluate integrals in Problem 1.

• Find an estimate of the area under the graph of between and above the -axis. Use four left endpoint rectangles. • Find an estimate of the area under the graph of between and above the -axis. Use four right endpoint rectangles. • Find an estimate of the area under the graph of between and . Use four left

? Find if . ? Find if and . ? A particle is moving with acceleration . Find the position function at time if and the velocity at zero, .

Let RI be the set of functions that are Riemann Integrable. Disprove with a counterexample or prove the following true. (a) f in RI implies |f| in RI (b) |f| in RI implies f in RI (c) f in RI and 0 < c <= |f(x)| forall x implies 1/f in RI (d) f in RI implies f^2 in RI (e) f^2 in RI implies f in RI (f) f^

1. In a mechanical system, the displacement of a body, s meters, is related to time t seconds, by the integral: t = (integral) [r/(g +ks)]ds where r, g and k are all constants. Determine an expression in terms of r and k, for the time taken by the body to travel a distance g/k meters from its initial position.

Please see the attached file for the fully formatted problems. Let A be an integral domain. For a prime ideal P C A and let S = A P be the complement of P in A. Observe that S is multiplicatively closed (WHY?) and form the subring A_p := S^-1 A of the field of fractions F of A. We regard A as a subring of A_p as usual - i.e.

Please see the attached file for the fully formatted problems.