Homework 2 Solutions Exercise 1. davissch/previous_classes/224_summer2011/224... · PDF...

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Transcript of Homework 2 Solutions Exercise 1. davissch/previous_classes/224_summer2011/224... · PDF...

  • Homework 2 Solutions

    Exercise 1. Use the Fundamental Theorem of Calculus to find the derivative of the function.

    1. g(x) = ∫ x 0

    √ 1 + 2t dt

    2. F (x) = ∫ 10 x tan θ dθ [Hint:

    ∫ 10 x f(θ) dθ = −

    ∫ x 10 f(θ) dθ]

    3. g(x) = ∫ 3x 2x

    u2−1 u2+1

    du

    Exercise 2. Find the average value of the function on the given interval.

    1. f(x) = x2, [−1, 1]

    2. g(x) = cosx, [0, π/2]

    1

  • Exercise 3. Evaluate the integral by making the given substitution.

    1. ∫ cos 3x dx, u = 3x

    2. ∫ x2 √ x3 + 1 dx, u = x3 + 1

    Exercise 4. Evaluate the integral using integration by parts with the indicated choices of u and dv.

    1. ∫ x lnx dx, u = lnx, dv = xdx

    2. ∫ θ sec2 θ dθ, u = θ, dv = sec2 θdθ

    2

  • Exercise 5. Decide which integration technique (substitution, int. by parts) is appro- priate and evaluate the integral.

    1. ∫ 2x(x2 + 3)4 dx

    2. ∫ x cos 5x dx

    3. ∫ (lnx)2

    x dx

    4. ∫ cos θ sin6 θ dθ

    5. ∫ 1/2 0 sin

    −1 x dx

    3

  • 6. ∫ t3et dt

    7. ∫ e2θ sin 3θ dθ

    8. ∫

    1+x 1+x2

    dx

    9. ∫ 2 0 (x− 1)

    25 dx

    10. ∫ sinπt dt

    4

  • 11. ∫ 1 0 ez+1 ez+z dz

    12. ∫ e4 e

    dx x √ lnx

    13. ∫ xe−x dx

    14. ∫ 1 0

    y e2y

    dy

    5

  • Exercise 6. If f is continuous and ∫ 4 0 f(x) dx = 10, find

    ∫ 2 0 f(2x) dx. If g(x) is contin-

    uous and ∫ 9 0 g(x) dx = 4, find

    ∫ 3 0 xg(x

    2) dx.

    Exercise 7. First make a substitution and then use integration by parts to evaluate the integral.

    1. ∫ x5 cos(x3) dx

    2. ∫ 4 1 e √ x dx

    6