Engineering Math II – Spring 2015 Quiz #1 Solutions
Problem 1 (5 pts) .
After an appropriate substitution, the integral alent to which of the following?
R
4
− 1 x
( x +7) 3 dx is equiv-
(a) R
11
(7 u
6
− 3 − u
− 2 ) du
(b) R
11
( u
6
− 2 − 7 u
− 3 ) du
(c) R
4
− 1
( u
− 2 − 7 u
− 3 ) du
(d)
R
4
− 1 xu
− 3 du
(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 u 3
, which is the same as ( u − 7) u
− 3
= u
− 2 − 7 u
− 3
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 3 t cos 2 tdt .
(a) − cos 5 t
5
− cos t + C
(b) cos 5 t
5
+ sin t + C
(c) cos 5 t
10
+ sin t
2
+ C
(d) − sin 5 t
10
− sin t
2
+ C
(e) − cos 5 t
10
− cos t
2
+ C
Solution.
The correct answer is choice (e). If we take A = 3 t and B = 2 t , then the integrand is sin A cos B . In terms of the identity given to us, this is equal to
B ) + sin( A − B ) = 1 sin 5 t + 1
1 sin( A +
2 sin t . Notice that when we take this integral we will get
2 2 something with cos 5 t and cos t in it, but there will be no term containing sin 5 t 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 respect to t is − 1
1 sin t . The integral of this function with
2 cos t , so choice (e) is the correct answer.
2

Quiz #1 Solutions 3
Problem 3 (5 pts) .
R sec θ tan θ
4+sec θ dθ =
(a)
1
4 ln | cos θ | + C
(b) sec θ
4 θ +ln | sec θ +tan θ |
+ 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, it is really a test to see if you remember the derivative of sec θ . Recall that d sec θ = dθ sec θ tan θ . This appears in the numerator of our integrand and immediately suggests that we should use u substitution. In this case we let u = 4 + sec θ . Then du = sec θ tan θdθ and the integral becomes R du u which is just ln | u | . Since u = 4 + sec θ , (c) is the right answer.
If you could not see which substitution to make, one strategy would be to take the derivatives of the answers and see which one gives you sec θ tan θ
. Doing it this way, you can
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 4
Problem 4 (5 pts) .
R cos
2
(2 x ) dx =
(a)
1
2 x +
1
4 sin(2 x ) + C
(b)
1
2 x +
1
8 sin(4 x ) + C
(c)
1
2 x − 1
8 sin(4 x ) + C
(d)
1
2 x − 1
4 sin(2 x ) + C
(e) sin
3
(2 x )
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 2 θ = 2 sin
2
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 cos
2
(2 x ), which from the double-angle formula is equal to
1+cos(4 x )
. This is the derivative
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.