Prob 1:
Several people forgot to use the 1/2 angle formula. Some integrated
cos 2u incorrectly.
Prob 2:
Some forgot the negative sign when substituting
u= cos 2x and taking the derivative.
Prob 3:
Some did not make a substitution for the exponent and incorrectly
integrated. Others made elementary arithmetic errors, e.g., thinking
e8 - e4 equals e4,
which is not true.
Prob 4:
The easiest way is to reduce the hyperbolic trig functions to
the definitions (in terms of e to some power), then multiply
out the expressions, and then integrate. Some people multiplied
incorrectly and others lost
a negative sign or a divisor of 3 in a couple of the expressions.
(Since the exponent in 3 of the 4 e-expressions was not
an "x-all by itself", an (implicit) u-substitution is
needed.) An alternative approach is to convert cosh 2x to something
in terms of cosh x or sinh x. Another alternative is to
reduce the hyperbolic trig functions and after multiplying the
expressions, combining those with similar powers and converting back
into a difference of two cosh expressions.
Prob 5:
The derivative of arctan x is 1/(1+x2).
But the derivative
of arctan u is 1/(1+u2) du/dx.
The derivative is not in the denominator. Some people had an
extra term in their answer.
Prob 6:
If the denominator were multiplied out, the highest power would be
4. Thus one needs to create FOUR partial fractions.
The first sum in the denominator factors to x(x+1), the second factors to 4(x2+1) which CANNOT be factored any further. Thus you should have 4 fractions to integrate: A/x, B/(x+1), C/(x2+1) and D(2x)/(x2+1), with the constant 4 being associated with any of them or multiplied outside all of them. Since the first sum in the denominator CAN be factored, it MUST be factored -- you cannot use the technique for non-factorable quadratic to handle the first sum.
Several people had different integrals, but misused the "ln u" rule and got the same answer! Anytime two DIFFERENT integrals give the SAME answer, you should ask whether something is wrong!
Several people correctly constructed the 4 partial fractions, but then didn't integrate them as the problem explicitly stated to do.
Prob 7:
Some people forgot that "solving" a differential equation means
coming up with a equation in x and y WITHOUT any derivative in it.
The values for x and y are used to determine the value of
C after
the integration (which is done after the "variables are separated").
Note that one should not "wait" to insert the
constant until after manipulating an equation
first -- this often leads to mathematical inconsistencies.
Numerous people forgot to include the absolute value signs around y after integration (in "ln|y|").
A number of people incorrectly attempted to "separate variables" resulting in x2 being in the numerator rather than the denominator.
Prob 8:
The function being integrated is always positive since all the powers are
even and the numerator is a sum. Thus the area under the curve (i.e., the
value of the integral) should be positive (if it exists).
A negative answer to "naive" integration should indicate something
is incorrect. In reality, this is an improper integral
which diverges. So to evaluate it, one must use the technique of
substituting b for the limit of integration at the problem point
and taking the appropropriate limit for b.
There were numerous simple algebra mistakes, e.g., people saying that x6=x2x3 or x2/x6 = 1/x3.
Prob 9:
From the many examples we did, 1-x2 demands the
substitution that x = sin u. Some people used all sorts
of substitutions other than this correct one.
A few people used a
"substitution" in which the new variable was the same as the
old variable, leading to confusion.
A few didn't express the final answer in terms of the original
variable x.
Prob 10:
Several people treated 1-x2 as if it were
"non-factorable." However, it IS factorable and it MUST be factored
before using the partial fractions techniques.
Several lost a negative sign when integrating 1/(1-x).
Several people neglected to find values for A and B.
Prob 11:
A couple of people lost a few points because they indicated
an answer that would take more work (i.e., it was not the "best/simplest").
E.g., c) should be done by simple substitution (#6) rather than
the secant substitution (#3) and f) should by done by simple
substitution (#6 with u = ln x) rather than by
parts (#7). A couple of people lost a few points because they did not
indicate what u (or dv) were when indicating options 6, 7
or 8.
final nfinal
246 69
245 69
240 67
238 66
236 65
229 62
228 62
227 62
227 62
223 60
221 59
220 59
219 58
215 57
215 57
214 56
205 53
199 50
198 50
197 50
193 48
186 45
177 42
168 38
163 36
151 31
144 28
121 19
111 15
83 4
MAXIMUM 250 100
x
x
x
x
x
x
x x x
x x x
x x x x
x x x x x x
x x x x x x x x
70- 90- 110- 130- 150- 170- 190- 210- 230-
89 109 129 149 169 189 209 229 250
(1) (0) (2) (1) (3) (2) (5) (11) (5)
This page is maintained by Dennis C. Smolarski, S.J.
dsmolarski at scu.edu
© Copyright 2009 Dennis C. Smolarski, SJ, All rights reserved.
Last changed: 19 March 2009.