### Evaluating the Integral

What is the integral of: Integral (2 --- 0) of x^2/ (4 +x^2)dx = ? A. 2 - ln(2) B. ln(2) - 1/2 C. 2 tan^2(2) + 4 ln[cos(2)] D. 2 ln[sec(2)]-sin^2(2) E. 2 - pi/2

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What is the integral of: Integral (2 --- 0) of x^2/ (4 +x^2)dx = ? A. 2 - ln(2) B. ln(2) - 1/2 C. 2 tan^2(2) + 4 ln[cos(2)] D. 2 ln[sec(2)]-sin^2(2) E. 2 - pi/2

What is the average value of y = (2x+1)1/2 over the interval [4,12]? A. √ 3 - 1/3 B. 2(√ 3 - 1/3) C. 4 D. 49/12 E. √ 17 Please explain solution in detail.

26. Let S be the closed region in the first quadrant of the xy-plane bounded by y = sin(pi x/2) and y = x for 0 ≤ x ≤ 1. What is the volume of the closed region in R3 obtained by revolving S about the x-axis? A. 2 - (pi /2) B. pi /6 C. pi /3 D. pi /2 E. (2pi )/3

25.∫(x-8)/(x2 - 4x) dx 6 ---> 8 A. -(47/576) B. 1/6 C. ln (8/9) D. ln 2 E. ln (32/9) A. -31 B. -19 C. 11 D. 30 E. 49 Please explain in detail. Thanks.

Please solve the attached integral substitution problem {also attached: multiple choice options} Thank you.

Please see the attached file for the fully formatted problems. 21. The region S is bounded by y = x2 - 2x + 3, y = 0, x = 0, and x = 9. Which of the following is the approximation to the area of S obtained by computing the sum of the areas of the 3 inscribed rectangles with bases [0,3], [3,6], and [6,9] (lower Riemann sum)?

20. Let S be the closed region in the first quadrant of the xy-plane bounded by y = 6x2, y = 0, x = 0, and x = 1. What is the volume of the solid obtained by revolving S about the line x = -1? A. 3x B. 7x C. 36x /5 D. 8x E. 56x /5

19. ∫0 to (pi/4) of x²cos x dx Please see attachment for full question.

18. Let F(x) = ∫0 to x^1/3 (√1+t^4) dt Then F'(0) = A. 0 B. 1/3 C. 2/3 D. 1 E. Does not exist. Please see attachment for full question.

17. If ∫0to1f(x)dx = -1 and ∫0to1g(x)dx= 1 then ∫1to0g(x)dx - ∫0to1 2f(x)dx = A. -3 B. -1 C. 0 D. 1 E. 3 Please see attachment for full question.

16. ∫0 to 3 x/(√x+1) dx = A. 3/8 B. 2/3 C. 3/2 D. 9/4 E. 8/3

15. ∫1 to ∞ 1/ (e^x +1)dx = A. ln (1 + e-1) B. - ln (1 + e-1) C. ln (1 + e) D. arctan (e1/2) E. does not exist Please see attachment

14. ∫dx/(x^3) dx -1 --> 2 A. (1/12)ln 8 B. 3/8 C. û(5/12) D. ln 8 E. does not exist

13. What is the average value of the function f(x) = x sin(x2) over the interval [2,4]? A. sin 4 + 2 sin 16 B. -sin 4 + 2 sin 16 C. (cos 4 - cos 16)/2 D. (cos 4 - cos 16)/4 E. (-cos 4 + cos 16)/4

12. What is the area of the closed region bounded by y = x2 - |x| and the x-axis, between x = -1 and x = 1? A. 1/12 B. 1/6 C. 1/3 D. 2/3 E. 5/6

Please see the attached file for the fully formatted problems. 11. ∫ x lnx dx 3 -->1 A. -2 + (9/4)ln 3 B. -4 + (9/2)ln 3 C. -(1/4) + (9/4)ln 3 D. -(5/2) + (9/2)ln 3 E. -2 + (9/2)ln 3 Please explain your answer in detail. Thanks.

Please see the attached file for the fully formatted problem. 10. ∫(x2 + 3x - 5)/x2 dx 6-->1 A. 5/6 - 3 ln 6 B. 61/6 + 3 ln 6 C. 5/6 + 3 ln 6 D. 17/6 + 3 ln 6 E. 17/6 - 3 ln 6

7. The closed region in the first quadrant bounded by the curves y = x3 and y = x(1/3) is rotated about the x-axis. What is the volume of the resulting solid? A. 1/2 B. 128x /455 C. 16x /35 D. x /2 E. 32x /35

What is the area of the closed region bounded by x = -1, x = 0, y = x2, and y = x3 ? A. 1/12 B. 1/6 C. 1/4 D. 5/12 E. 7/12

2. What is the average value of xe^x2 on the interval [2, 4]? A. (e16 - e4)/4 B. (e16 - e4)/2 C. (2e16 - e4)/2 D. 2e16 - e4 E. 2e16 + e4

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.)

Integrate the function (1/θ)^2 e^((-x1-x2)/ θ)) from 0 to 2 ln 2- x2 (for dx1) and from 0 to 2 ln 2 (for dx2).

∫(1 + tanθ)^5sec^2θ dθ

∫ √x sin(1 + x^3/2)dx (Corrected)

∫sec2θ tan2θ dθ

∫y^3 √(2y^4 -1)

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))