Download Quiz 6 - UTK Math

Quiz 6. Time: 15 minutes
KEY
Name:
1
Math 142, Calculus II, Fall 2016
Section:
ID#:
(i) (2pts) Integration by parts formula is:
Z
Z
Z
Z
Z
Z
u dv = uv − v du.
A.
u dv = uv − u du.
B.
u dv = uv − v dv. C.
Z
Z
D.
u dv = uv + v du.
Z
(ii) (2pts) If u = x and dv = sin(x) dx, then the integral
A. x cos(x) + sin(x) + C.
D. −x cos(x) + sin(x) + C
x sin(x) dx =
B. x cos(x) − sin(x) + C.
C. −x cos(x) − sin(x) + C
Solution:
u=x
,
du = dx
dv = sin(x) dx
Z
, v = sin(x) dx = − cos(x)
Thus,
Z
Z
x sin(x) dx = −x cos(x) −
Z
− cos(x) dx = −x cos(x) +
cos(x) dx = −x cos(x) + sin(x) + C.
We can do it using Table Method as follows:
D
I
x
+
sin(x)
1
-
− cos(x)
0
− sin(x)
J
Z
(iii) (1pt) Consider the integral
x3 ln(x) dx. This integral can be calculated only by using integration by parts.
Which of the following is the best choice for u and dv?
A. u = x3 and dv = ln(x) dx.
B. u = ln(x) and dv = x3 dx
C. u = x2 and dv = x ln(x) dx
D. u = x ln(x) and dv = x2 dx
Z
(iv) (1pt) Consider the integral
x e5x+2 dx. This integral can be calculated only by using integration by parts.
Which of the following is the best choice for u and dv?
A. u = x and dv = e5x+2 dx.
B. u = e5x+2 and dv = x dx
C. u = 5x + 2 and dv = x dx
5x+2
D. u = e
and dv = dx
Z √
(v) Consider the integral e x dx. This integral can be calculated by using substitution first then integration by
parts. Answer the following.
√
(v1) (1pt) Using the substitution w = x = x1/2 , the integral can be written as
Z
Z
Z
Z
2 w ew dw
D.
2 ew dw
A.
ew dw
B.
w ew dw C.
Quiz 6. Time: 15 minutes
2
Math 142, Calculus II, Fall 2016
(v2) (1pt) Using Zintegration by parts calculate the integral you chose in part (v1). Now replacing w by
√
the integral
e x dx =
√
A. 2 e
x
√
( x − 1) + C.
√
B. e
x
√
( x − 1) + C.
√
C. e
x
√
+ C.
D. 2 e
x
√
x,
+ C.
Solution:
u = 2w
du = 2dw
,
dv = ew dw
Z
, v = ew dw = ew
Thus,
Z
w
Z
w
2 w e dw = 2w e −
2 ew dw = 2w ew − 2 ew + C = 2ew (w − 1) + C.
We can do it using Table Method as follows:
D
I
2w
+
ew
2
-
ew
ew
0
Replacing w by
√
x, we have
√
Z
e
x
√
dx = 2e
x
√
( x − 1) + C.
J
Q: (2pts) Do you like math? Why or why not? What is the most you like about the class on Tuesday?