Homework 2Solutions

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

1. g(x) =∫ x0

√1 + 2t dt

2. F (x) =∫ 10x tan θ dθ [Hint:

∫ 10x f(θ) dθ = −

∫ x10 f(θ) dθ]

3. g(x) =∫ 3x2x

u2−1u2+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 choicesof 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.∫ 1/20 sin−1 x dx

3

6.∫t3et dt

7.∫e2θ sin 3θ dθ

8.∫

1+x1+x2

dx

9.∫ 20 (x− 1)25 dx

10.∫sinπt dt

4

11.∫ 10ez+1ez+z dz

12.∫ e4e

dxx√lnx

13.∫xe−x dx

14.∫ 10

ye2y

dy

5

Exercise 6. If f is continuous and∫ 40 f(x) dx = 10, find

∫ 20 f(2x) dx. If g(x) is contin-

uous and∫ 90 g(x) dx = 4, find

∫ 30 xg(x

2) dx.

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

1.∫x5 cos(x3) dx

2.∫ 41 e√x dx

6