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