Math 165 - Assignment #6
Name:
Due 7/30/2013
SHOW YOUR WORK!
This assignment covers sections 1 -4 from chapter 4. Make sure you justify your answer clearly! Simply writing the
final result does not mean you receive full credit. You must also show you understand the procedure.
1.- Suppose that
1
Z
Z
f (x) dx = 1,
0
2
Z
1
1
Z
g(x) dx = −1,
f (x) dx = 2,
0
2
g(x) dx = −2
0
Use the linearity of the definite integral. Use the linearity of the definite integral to find the following.
√
Z 2√
2
(a)
3 f (x) +
g(x) dx
12345
1
+
π
0
Z
(b)
1
π f (x) +
√
π g(x) dx
2
2.- Below your are given a function f (x). Find f 0 (x) by using the First Fundamental Theorem of Calculus.
Z cos(x)
(a) f (x) =
t2 dt
sin(x)
x
Z
x2 t dt
(b) f (x) =
0
3.- Use the Second Fundamental Theorem of Calculus to evaluate the following functions. Careful! You may have to
use the method of u-substitution.
Z π/2
2 sin(x) dx
(a)
−π/2
Z
1
x4/3 − 2x1/3 dx
(b)
0
Z
(c)
1
3
x1/2
1
dx
+ x3/2
Z
(d)
π/2
x3 sin2 (x4 ) cos(x4 ) dx
0
4.- Consider the function f (x) = sec2 (x) tan(x).
R
(a) Solve the indefinite integral f (x) dx using the u-substitution u = sec(x).
(b) Solve the indefinite integral
R
f (x) dx using the u-substitution u = tan(x).
(c) Assuming you evaluated (a) and (b) correctly, explain why both answers are the same.
5.- Use the method of u-substitution to find the following indefinite integrals.
Z
√
3
(a)
4x − 2 dx
Z
(b)
Z
(c)
x4 sin(5x5 + 1)
x3
p
x2 + 1 dt
p
3
cos(5x5 + 1) dx