Description
Problem 1
(10 points)
Use substitution to evaluate the following integrals.
a) R
sin(π/x2
)
x
3
dx (2 points)
b) R
2 ln x
x
dx (2 points)
Use integration by parts to evaluate the following integrals.
c) R
cos(x) ln(sin x)dx (3 points)
d) R π/2
0
x cos(x) sin(x)dx (3 points)
Problem 2
(10 points)
a) Prove the reduction formula
Z
cosn
(x) dx =
1
n
cosn−1
(x) sin(x) + n − 1
n
Z
cosn−2
(x) dx .
(5 points)
Hint: Use integration by parts and the fact that cos2
(x) + sin2
(x) = 1.
b) Suppose that f : R → R continuous and odd, i.e., satisfies −f(x) = f(−x). Show that
Z a
−a
f(x) dx = 0 .
(5 points)
Problem 3
(10 points + 5 bonus points)
Using 2a)
a) Evaluate Z
cos2
(x) dx. (4 points)
b) Evaluate Z
cos3
(x) dx. (3 points)
c) Evaluate Z 2π
0
cos5
(x) dx. (3 points)
Bonus:
We will cover this on Tuesday, 26.11:
Find the area between the curves x = 1 − y
2 and y = −2x − 1. (5 bonus points)
Hint: At the end, integrate with respect to y not x.
