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Suppose that we are using substitution to find an antiderivative. If, after making

Question : Suppose that we are using substitution to find an antiderivative. If, after making : 2151770

Find the Integral.

21) ∫(3x2)(4)√(10 + 2x3)

A) (12/5)(10 + 2x3)5/4 + C

B) (2/5)(10 + 2x3)5/4 + C

C) - 2(10 + 2x3)-3/4 + C

D) 3(10 + 2x3)5/4 + C

22) ∫(x6√(x7 + 3)dx)

A) (1/14√(x7 + 3)) + C

B) (2/21)x7(x7 + 3)3/2 + C

C) (2/3)(x7 + 3)3/2 + C

D) (2/21)(x7 + 3)3/2 + C

23) ∫((t3/(5)√(2 + t4))dt)

A) (5/16)t4(2 + t4)4/5 + C

B) (1/16(2 + t4)4) + C

C) (5/24)(2 + t4)4/5 + C

D) (5/16)(2 + t4)4/5 + C

24) ∫((x2 + 8x/(x + 4)2)dx)

A) (16/x + 4) + C

B) x + (16/x + 4) + C

C) x + (4/x + 4) + C

D) x + (32/(x + 4)3) + C

25) ∫((t3 + 1/t4 + 4t + 7)dt)

A) (ln|t4 + 4t + 7|/4) + C

B) - (4/(t4 + 4t + 7)2) + C

C) - (1/4(t4 + 4t + 7)2) + C

D) 4ln|t4 + 4t + 7|+ C

26) ∫((6x/(x + 4)4)dx)

A) 6ln|x + 4|3 - 24ln|x + 4|4 + C

B) 6ln|x + 4|+ C

C) - (4x/(x + 4)3) + C

D) - (3/(x + 4)2) + (8/(x + 4)3) + C

27) ∫((19/2 + 5y)dy)

A) 19ln|2 + 5y|+ C

B) (18/5)ln|2 + 5y|+ C

C) 18ln|2 + 5y|+ C

D) (19/5)ln|2 + 5y|+ C

28) ∫(((lnx)6/x))dx

A) ((lnx)5/5) + C

B) ((lnx)7/7) + C

C) (lnx)7 + C

D) ((lnx)7/7x) + C

29) ∫((1/x(lnx5)))dx

A) (1/5)ln|lnx5|+ C

B) (1/5)lnx5 + C

C)lnx5 + C

D)ln|lnx5|+ C

30) ∫((lnx6/x)dx)

A) (1/lnx6) + C

B) (1/12)(lnx6)2 + C

C) (1/6)(lnx6)2 + C

D) (1/2)(lnx6)2 + C

31) ∫(((lnx)6/x)dx)

A) ((lnx)7/x) + C

B) 6(lnx)5 + C

C) ((lnx)7/7x ) + C

D) ((lnx)7/7) + C

32) ∫((1/x(lnx)5)dx)

A) - (1/6(lnx)6) + C

B) (1/x(lnx)6) + C

C) - (1/4(lnx)4) + C

D) - (1/4x(lnx)4) + C

33) ∫((t4ln(t5 + 9) /t5 + 9)dt)

A) [ln(t5 + 9)]2 + C

B) ([ln(t5 + 9)]2/10) + C

C) ((lnt)2/10) + C

D) (t5ln(t5 + 9) /t5 + 9) + C

34) ∫(((6 +lnx)4/x)dx)

A) ((6 +lnx)5/5) + C

B) ((6 +lnx)5/5x2) + C

C) ((6 +lnx)5/5x) + C

D) 5x2(6 +lnx)5 + C

35) ∫((1/3x(lnx)))dx

A)ln|3lnx|+ C

B) (ln|x|+ln|lnx|/3) + C

C) 3ln|lnx|+ C

D) (ln|lnx|/3) + C

36) ∫((log2x/x))

A) ((log2x)2/2) + C

B) ((lnx)(log2x)2/2) + C

C) ((log2x)2/2ln2) + C

D) ((ln2)(log2x)2/2) + C

37) ∫(((log9(3x - 3))2 /3x - 3))

A) ((log9(3x - 3))3/9)

B) ((ln9)(log9(3x - 3))3/9)

C) ((log9(3x - 3))3/9ln9)

D) ((ln9)(log9(3x - 3))3/3)

Solve the problem.

38) The rate of expenditure for maintenance of a particular machine is given by M'(x) = 9x√(x2 + 5), where x is time measured in years. Total maintenance costs through the second year are $99. Find the total maintenance function.

A) M(x) = 3(x2 + 5)3/2 + 18

B) M(x) = 9(x2 + 5)3/2 + 90

C) M(x) = 9(x2 + 5)3/2 + 18

D) M(x) = 3(x2 + 5)3/2 + 90

39) The rate of growth of the profit (in millions) from an invention is approximated by P'(x) = xe-x2, where x represents time measured in years. The total profit in year 1 that the invention is in operation is $30,000. Find the total profit function. Round to three decimal places where appropriate.

A) P(x) = -0.5e-x^2 + 214,000

B) P(x) = -0.5e-x^2 - 0.214

C) P(x) = -0.5e-x^2 + 0.214

D) P(x) = -0.5e-x^2 - 214,000

40) The work W (in joules) done by a force F (in newtons) moving an object through a distance x (in meters) is given by W = ∫(Fdx). Find a formula for W, if F = kx and k is a constant.

A) W = (kx/2) + C

B) W = k + C

C) W = kx2 + C

D) W = (kx2/2) + C

41) A company has found that the marginal cost of a new production line (in thousands) is C'(x) = (9/x + e), where x is the number of years the line is in use. Find the total cost function for the production line (in thousands). The fixed cost is $20,000.

A) C(x) = (ln(x + e)/9) + 11

B) C(x) = 9ln(x + e) + 20

C) C(x) = (ln(x + e)/9) + 20

D) C(x) = 9ln(x + e) + 11

42) The current (in amperes) in an inductor of constant inductance L (in henries) is given by i = (1/L)∫(Vdt), where V is the voltage (in volts) and t is the time (in seconds). Find a formula for i, if V = 9t(t2 - 6).

A) i = (1/L)((9/4)t4 - 27t2) + C

B) i = L((9/4)t4 - 27t2) + C

C) i = (1/L)((9/4)t4 - 54t) + C

D) i = (1/L)((9/4)t4 - 54t2) + C

Answer the question, concerning the use of substitution in integration.

43) If we decide to use u = x + e, then which of the following are correct?

i) du =dx

ii) x = u - e

iii) du = edx

iv) dx = edu

A) Both ii and iii

B) Only i

C) Both i and iv

D) Both i and ii

44) If we decide to use u = x2 + e, then which of the following are correct?

i) du = 2xdx

ii) x2 = u - e

iii) du = 2exdx

iv) du = (2x/3)dx

A) Both ii and iii

B) Both i and iv

C) Only i

D) Both i and ii

45) If we use u = x2 as a substitution to find ∫(xex^2dx,) then which of the following would be a correct result?

A) ∫(du)

B) ∫(2eudu)

C) ∫((eu/2)du)

D) None of the above

46) If we decide to use u = ex2, then which of the following are correct?

i) du = 2edx

ii) x2 = (u/e)

iii) du = 2exdx

iv) du = ((2x/e))dx

A) Both ii and iii

B) Both i and iii

C) Only iii

D) None of the above

47) If we use u = x2 + 2 as a substitution to find ∫((x2 + 2)dx,) then which of the following would be a correct result?

A) ∫((u - 2)du)

B) ∫(udu)

C) ∫((u/2)du)

D) None of the above

48) If we use u = x + e as a substitution to find ∫((x + e)ndx,) then which of the following would be a correct result?

A) ∫(un+1du)

B) ∫(undu)

C) ∫(undx)

D) None of the above

49) Suppose that we are using substitution to find an antiderivative. If, after making the substitution, we find that there is still an x-term left in the integrand, what should we do?

A) This would never happen if we made the correct substitution.

B) Go back to the equation relating x and u, solve for x, and substitute in the integrand.

C) Use an alternative method, because substitution will never give the antiderivative.

D) None of the above

Solution
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