# Math Test #1 Review Sheet

21-122 Integration, Differential Equations and
Approximation Test #1 Review Sheet |
September 22, 2005 |

Test #1 will be held in the lecture on Monday, September
26, 2005. It will cover §7.1, §7.2, §7.3, §7.4, §7.7, §7.8, and

§8.1. You cannot use a calculator, books, or notes. The test will be 50 minutes
long.

This review sheet is intended to help you prepare for the test . It states pretty
much what we have done in a nutshell,

but this list is not exhaustive. Please be sure you are familiar with all the
concepts listed here. If you’re having trouble,

be sure to stop in at Professor Mackey’s or your TA’s office hours.

There will be a review session held on Sunday, September 25, 2005 from 3:00pm
until 4:00pm in Porter Hall 100 (the

large lecture hall). The session will be led by Sunil Raman, the TA for Section
H. I highly recommend attending this,

since it will help answer any last-minute questions you may have.

I attached a practice test for you. All but one of the questions came directly
from an actual test that was given in the

Fall 2002 semester. Hopefully it will help you study.

**§7.1 Integration by Parts**

• Integration by parts forumulas:

–Indefinite integrals:

– Definite integrals:

• You want the integral on the right hand side to be
easier to solve than the original integral. If it’s harder, chances

are you made the wrong choice for u and dv.

• If the integral on the right hand side is exactly equal to the original, solve
for it.

• How to choose u: **LIATE**

**L** ogarithmic

**I **nverse trigonometric functions

**A** lgebraic (i.e. polynomials, terms under square roots, etc.)

**T** rigonometric functions

**E** xponentials

• In all cases, you want u to be easy to differentiate and dv to be easy to
integrate.

**§7.2 Trigonometric Integrals
**

• You need to know trigonometric identities to do well in this section.

• If you have

– If n is odd, split off a copy of cosx, let cos^{2}
x = 1−sin^{2} x, then let u = sin x.

– If m is odd, split off a copy of sinx, let sin^{2} x = 1−cos^{2}
x, then let u = cosx.

– If both powers are even , let

• If you have

– If n is even, let sec^{2} x = 1+tan^{2} x, then let u = tan x.

– If m is odd, split off a copy of secx tan x, let tan^{2} x = sec^{2}
x−1, then let u = sec x.

• Remember that .

• Don’t be afraid to multiply the integrand by factors like
or . You
need to do this when integrating

secx, for example.

• You do** NOT** need to know the formulas in the box on page 487.

**§7.3 Trigonometric Substitution
**

• You need to know trigonometric identities to do well in this section, too.

• What to substitute:

• Remember to draw a triangle to figure out the values for
the other trigonometric functions are.

• Expression not in one of the forms above? Try completing the square and
factoring .

• Remember to change the limits if doing a definite integral! If you don’t,
you’ll get the wrong answer.

**§7.4 Integration by Partial Fractions
**

• Before trying to use partial fractions, try to simplify : if deg(numerator) ≥ deg(denominator), divide using

polynomial long division.

• Factor all terms in the denominator if possible.

• 4 cases to consider and what to do:

1. Distinct Linear Factors : Split into form

2. Repeated Linear Factors : Split into form

3. Distinct Quadratic Factors : Split into form

4. Repeated Quadratic Factors: Split into form

•

**NOTE:**The quadratic factors in the latter 2 cases

**must be irreducible**. This will not work if the quadratic

factors are reducible.

• Often you’ll have mixtures of these cases. Just remember to treat each factor in the denominator separately . For

example,

**§7.7 Approximate Integration
**

• You do

**NOT**need to memorize the formulas for the Midpoint Rule, the Trapezoid Rule, and Simpson’s Rule.

You also do not need to memorize the error formulas. All formulas in this section will be given to you on the

test should you need them.

• You

**DO**need to how these formulas arise geometrically.

• Remember that for f (x), if is an underestimate of , then is an overestimate of , and vice

versa. A lot of people missed this on Homework 3, and you need to know this.

**§7.8 Improper Integrals**

• How to set them up:

• In the setup above, pick s to be something convenient,
like 0.

• If f (x) has a vertical asymptote, split the integral there, and take left and
right hand limits.

• If you split an integral and one portion diverges, **stop**. The integral
diverges.

• Remember to carry the limits all the way through. Not doing this will cause
you to lose points.

**§8.1 Arc Length**

• Arc length of f (x) from (a, f (a)) to (b, f (b)) is
.

• You may need to use Simpson’s Rule or other approximate integration methods if
the integrand gets ugly.

21-122 Integration, Differential Equations and
ApproximationPractice Test |
September 22, 2005 |

This practice test is intended to help you study for Test
#1, to be given on Monday, September 26, 2005. All but

one of these questions came from an actual test which was given in the Fall 2002
semester. You should take this test

in a quiet area with all books, notes, and calculators put away and out of
sight. You have 50 minutes. There is a total

of 104 points (4 bonus points). The point value is indicated in brackets [ ].

**NOTE: **This test is (hopefully) more difficult than what you will be given
on Monday.

[16] **QUESTION 1** Evaluate the indefinite integral
Show all your work.

[16] **QUESTION 2** Make a substitution to express the
integrand below as a rational function and evaluate the resulting

integral. Show all your work.

[20] **QUESTION 3** Set up and evaluate an expression
involving a definite integral to find the formula for the area of a

sector of a circle of radius r and central angle θ. Assume
and consider a circle whose center is at the
origin

as in the diagram below. Show all your work.

[12] **QUESTION 4** Evaluate the indefinite integral
. Show all your work.

[8] **QUESTION 5** Determine if the improper integral
converges or diverges. Show all your work.

**QUESTION 6** Consider the definite integral
.

[10] **PART A** Using the Trapezoid Rule formula ,
write down (but

do not evaluate) an expression for .

[10] **PART B** Show how to find the smallest number K
such that on [0,1] and then use the error
formula

to find a value of n for which the error
using is within 0.01.
(Note to Section D: This was the

question I meant to ask you in recitation last week.)

[12] **QUESTION 7** Find the length of the curve y =
ln(sec x) such that . Show all your work.

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