Engineering Math II – Spring 2015
Problem 1 (5 pts). After an appropriate substitution, the integral
alent to which of the following?
R 11
(a) 6 (7u−3 − u−2 )du
R 11
(b) 6 (u−2 − 7u−3 )du
R4
(c) −1 (u−2 − 7u−3 )du
R4
(d) −1 xu−3 du
Quiz #1 Solutions
R4
x
dx
−1 (x+7)3
is equiv-
(e) None of the above
Solution. The correct answer is choice (b). Here’s one way to work this out. Your first
instinct should be to make the substitution u = x + 7. Then du = dx and we can write
x = u−7. The integrand then becomes u−7
, which is the same as (u−7)u−3 = u−2 −7u−3 .
u3
This essentially eliminates answers (a) and (d) from our choices, leaving either (b), (c) or
(e). The only difference between (b) and (c) is the limits of integration. Since u = x + 7,
the limits definitely change, so that choice (c) can be eliminated. This leaves (b) and
(e) as possible answers. We only need to check that the limits of integration in (b) are
correct. Our substitution was u = x + 7, so u = 6 when x = −1, and u = 11 when x = 4.
These match the limits of integration in choice (b), making this the correct answer.
Quiz #1 Solutions
2
Problem 2 (5 pts). Use the trigonometric
identity 2 sin A cos B = sin(A+B)+sin(A−B)
R
to evaluate the indefinite integral sin 3t cos 2tdt.
(a) − cos5 5t − cos t + C
(b)
cos 5t
5
+ sin t + C
(c)
cos 5t
10
+
sin t
2
+C
(d) − sin105t −
sin t
2
+C
(e) − cos105t −
cos t
2
+C
Solution. The correct answer is choice (e). If we take A = 3t and B = 2t, then the
integrand is sin A cos B. In terms of the identity given to us, this is equal to 12 sin(A +
B) + sin(A − B) = 21 sin 5t + 12 sin t. Notice that when we take this integral we will get
something with cos 5t and cos t in it, but there will be no term containing sin 5t or sin t.
This eliminates choices (b), (c) and (d), leaving just (a) and (e) as possible answers. To
determine which one it is, just look at the term 12 sin t. The integral of this function with
respect to t is − 21 cos t, so choice (e) is the correct answer.
Quiz #1 Solutions
Problem 3 (5 pts).
(a)
1
4
(b)
sec θ
4θ+ln | sec θ+tan θ|
3
R
sec θ tan θ
dθ
4+sec θ
=
ln | cos θ| + C
+C
(c) ln |4 + sec θ| + C
(d) ln |4 + tan θ| + C
(e)
1
4
ln | sec θ| + C
Solution. The correct answer is (c). Although this problem may look terrifying at first,
d
sec θ =
it is really a test to see if you remember the derivative of sec θ. Recall that dθ
sec θ tan θ. This appears in the numerator of our integrand and immediately suggests that
we should use u substitution.
R In this case we let u = 4 + sec θ. Then du = sec θ tan θdθ
which is just ln |u|. Since u = 4 + sec θ, (c) is the right
and the integral becomes du
u
answer.
If you could not see which substitution to make, one strategy would be to take the
θ tan θ
. Doing it this way, you can
derivatives of the answers and see which one gives you sec
4+sec θ
immediately eliminate choices (a), (b) and (e), since there is no way for those functions
to have the derivative we want. At this point you have a 50/50 chance of getting the
answer right, even if you don’t know how to take the integral or remember the derivative
of sec θ.
Quiz #1 Solutions
Problem 4 (5 pts).
4
R
cos2 (2x)dx =
(a)
1
x
2
+ 41 sin(2x) + C
(b)
1
x
2
+ 81 sin(4x) + C
(c)
1
x
2
− 81 sin(4x) + C
(d)
1
x
2
− 41 sin(2x) + C
(e)
sin3 (2x)
3
+C
Solution. The correct answer is (b). In all fairness, at the time you took this quiz you had
not seen how to take an integral like this one. This was meant to be a thought-provoker.
In order to take this integral, you need to use the double-angle formula cos 2θ = 2 sin2 θ−1.
I do not expect you to have this formula memorized, and the intention was that some
of you may actually discover that you need this identity on your own. In light of this,
full credit for this problem will be given to those who attempted to solve it and showed
sufficient work, regardless of whether or not their answer is correct.
If you know the identity, there is a really easy way to determine the correct solution:
just take the derivatives of the answer choices. This immediately eliminates choice (e).
[Notice that at this point, even if you don’t know the identity and don’t know how to take
the integral, just taking the derivatives and eliminating (e) raises your chances of getting
the correct answer by guessing from 20% to 25%.] The derivative should be equal to
. This is the derivative
cos2 (2x), which from the double-angle formula is equal to 1+cos(4x)
2
of choice (b), making that the correct answer. Now that you’ve seen that trick, you
should “check” our answer by evaluating the integral using the double-angle formula.
It is difficult to know what formulas you will be expected to have memorized for the
exam. Dr. Reihani is your best source for this information. That said, you may wish to
review double- and half-angle formulas and other common trig identities as they can be
extremely useful for simplifying an integral.