Math 230 Exam 2 Review Sheet Spring 2007

Math 230 Exam 2 Review Sheet Spring 2007
The following basic trig identies will be the only formulas provided for you. They will be provided in the
following forms:
sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y − sin x sin y
tan x + tan y
tan(x + y) =
1 − tan x tan y
sin 2 1 − cos 2x
x =
2
sin x cos x = 1 2 sin(2x)
sin A sin B = 1 [cos(A − B) − cos(A + B)]
2 cos
sin(x − y) = sin x cos y − cos x sin y
cos(x − y) = cos x cos y + sin x sin y
tan x − tan y
tan(x − y) =
1 + tan x tan y
cos 2 1 + cos 2x
x =
2
sin A cos B = 1 [sin(A − B) + sin(A + B)]
2 A cos B = 1 [cos(A − B) + cos(A + B)]
2
The ∫ following are forms∫that you should ∫be able to recognize ∫ (these will not be provided for you):
1
x n dx (n ≠ −1)
x dx e x dx a x dx
∫
∫
∫
∫
sin xdx
cos xdx sec 2 xdx csc 2 xdx
∫
∫
∫
∫
sec x tan xdx csc x cot xdx tan xdx cot xdx
∫
∫
∫
∫
1
sec xdx
csc xdx
x 2 + a 2 dx 1
√
∫
a2 − x dx 2
1
x √ x 2 − 1 dx
Definite integrals must be completely evaluated (i.e. you must be able to plug in standard angles for
trig and inverse trig functions) in order to receive full credit. You will not be told what technique to apply
to an integral. You need to make sure that you can do each of the techniques correctly, and you must do
enough problems to be able to recognize a viable approach to an integral. The first part of this review breaks
down each of the sections with some typical integrals from those sections. The second part of the review is
a strategy for integration and some integrals for you to practice identifying techniques to use.
1. L’Hospital’s Rule: Know when to apply it and verification of all conditions must be shown in order to
receive full credit. You must be able to state the conditions necessary for application of the rule.
x + tan 2x (ln x) 3 e x (
− 1
1
lim
lim
x→0 x − tan 2x x→∞ x 2 lim
lim
x→0 sin x x→1 ln x − 1 )
x − 1
lim
x→0
sin x
x 3
sin x
lim
x→0 e x
lim (cos x)1/x2 lim
x→∞
→∞
(1 − 5 x
2. Inverse Trig Functions: No integral or differentiation formulas will be provided.
(a) Differentiate the following:
y = √ tan −1 x 2 y = e sec−1 t
y = x cos −1 x − √ 1 − x 2
y = sin −1 (2x 2 + e x2 ) y = sin(cos −1 (2x − 1)) y = ln(2x 2 ) tan −1 (x 2 + x)
(b) Evaluate the following:
∫ 1/2
0
∫ √ 3
dx
√
1 − x
2
∫
6
1 1 + x 2 dx ∫
3. Integration by parts:
∫ x + 4
x 2 + 4 dx
e 2x
√
1 − e
4x dx
∫
udv = uv −
∫ tan −1 x
1 + x 2 dx
∫
e x
e 2x + 1 dx
vdu
) x

∫
∫
e −x (x 3 + 7x + 2)dx
x 3 tan −1 xdx
∫
∫
x cos xdx
(ln x) 2 dx
∫
x5 x dx
∫
x 5 e x2 dx
4. Trigonometric Integrals:
∫
(a) the form sin m x cos n xdx
i. cos x power odd - save one cos x factor
ii. sin x power odd - save one sin x factor
iii. both cos x and sin x have odd powers - can save your favorite
iv. both cos x and sin x have even powers - use the half-angle identites to convert to an easier
form
∫
(b) the form tan m x sec n xdx
i. sec x power even - save one sec 2 x factor
ii. tan x power is odd - save one sec x tan x factor
∫
∫
∫
(c) the forms sin(mx) cos(nx)dx, sin(mx) sin(nx)dx, and cos(mx) cos(nx)dx require the appropriate
conversions using formulas for products of trig functions.
∫ π/2
∫
∫ π/4
cos 4 xdx tan 4 xdx
sec 6 xdx
∫0
∫ 0
sec
sin 6 x cos 3 2 ∫
x
xdx
cot x dx sin(5x) sin(2x)dx
5. Trig Substitution: can be used when √ a 2 − x 2 , √ a 2 + x 2 , √ x 2 − a 2 show up in an integral.
∫ 3
∫
dx
1 √ ∫
dx
√ x2 + 1dx √
0 9 + x
2
∫
∫ 0
x2 − 6x + 13
dx
√ e t√ ∫ 3
9 − e 2t dt x √ 9 − x 2 dx
9x2 + 6x − 8
0
6. Partial Fractions: you will be asked to express a complicated rational function in partial fraction
terms without solving for the constants or evaluating the integral. You will also be asked to solve one
completely. Remember that it is only applicable when the degree of the denominator is larger than
the degree of the numerator and that the denominator must be completely factored into irreducible
quadratics and linear factors.
(a) Express the following in a partial fraction decomposition without solving for the terms.
i.
ii.
4x 2 − 2x + 9
x 3 (x + 1) 2 (x − 1)(x 2 − 4) 2 (x 2 + 2)(x 2 + 3) 3
6x 2 − x + 13
x 2 (x − 1)(x + 3) 2 (x 2 + 7)(x 2 − 7) 2 (x 2 + x + 1) 3
(b) Solve the following completely
∫ x 3 + x 2 ∫
− 12x + 1 x 3 − 2x 2 + x + 1
x 2 dx
+ x − 12
x 4 + 5x 2 dx
∫ + 4
x 2 ∫
+ 1
1
x 2 − x dx
Strategy for integration:
1. Simplify the Integrand if Possible:
(a) distribution of terms
(b) splitting of fractions
0
∫
1
x 3 + x 2 − 2x dx
x 2 − x + 6
x 3 + 3x dx ∫ x 3 + x 2 + 2x + 1
(x 2 + 1)(x 2 + 2) dx

(c) trig identities
2. Look for an obvious substitution. This is why you need to KNOW the list of integrals at the beginning
of this sheet.
3. Classify the integrand according to its form
(a) Powers of trig functions? – trigonometric integrals good place to start
(b) Rational function? – partial fractions a good place to start
(c) Product of two functions? – parts a good place to start
(d) Radicals? – perhaps a trig sub will get you there
4. Still no answer? Try again.
(a) Did you miss a substitution?
(b) Will parts work? Although it is usually used on products of functions, it can be used on single
functions.
(c) Can you manipulate the integrand? i.e. rationalize the denominator, trig identities, etc.
(d) Have you seen the problem before? Can you relate it to a previous homework problem or example
from class?
(e) Sometimes more than one method is required. You might need to do substitutions within parts
or some other combination of techniques.
Integrate ∫ the following by any method or combination of methods.
e x
∫
1.
1 + e x dx 2. e x
∫
1 + e 2x dx 3. e √x dx
∫
∫
∫
1
4.
1 + e 2x dx 5. t 3 e −t2 dt 6. sin(ln x)dx
∫
∫
∫
sin(2x)
7.
1 + sin 2 x dx 8. sin 3 x ln(cos x)dx 9. (sin x) e (cos x) 3 dx
∫
∫
∫
10. sec 3 x tan 3 xdx 11. (ln x) 2 ln x
dx 12.
x √ 1 + ln x dx
∫
13. tan −1 ( √ ∫ sin −1 ∫
x
1
t)dt 14.
x 2 dx 15.
∫
∫
∫ sec x + tan x dx
1
tan −1
16. sin(3θ) cos(θ)dθ 17.
1 − sin x dx 18. (e x )
e x dx
∫
x 2
∫ sin 2
19.
(1 + x 2 ) 2 dx 20. x − cos 2 ∫
x
√3
dx 21. − x2 dx
∫
∫ √ cos x
∫
x
3 − x
22. √ dx 23. 2
1
dx 24.
3 − x
2 x
(x 2 − 4x + 4)(x 2 − 4x + 5) dx
∫
∫
∫
1
25.
(a sin x + b cos x) 2 dx 26. 1
π/2
a 2 sin 2 x + b 2 cos 2 x dx 27. sin x
0 1 + cos x + sin x dx
(a ≠
∫
0) (ab
∫
≠ 0)
1
28. √
x2 + x dx
29. x 4 ∫
+ 1
x(x 2 + 1) 2 dx 30. 4x 5 − 1
∫
∫
(x 5 + x + 1) 2 dx
1
31.
x − √ x dx
32. sin x sin 2x sin 3xdx

Math 230 Exam 2 Spring 2002
1. Find the derivatives of the following functions:
(a) y = ln(2x) tan −1 (x 2 )
(b) y = sin(cos −1 (2x − 1))
2. Find the following limits:
(a) lim
x→0
1 − cos 2x
x sin s
(b)
lim
x→∞ (1 − 5 x )x
3. Evaluate the following integrals:
∫
(a) (4 − x)e −3x dx
∫
(b) x 2 tan −1 xdx
(c)
∫
e
cos −1 x
√
1 − x
2 dx
(d)
(e)
(f)
(g)
∫ 1
∫
∫
0
x 2
√
4 − x
2 dx
sin 3 x cos 4 xdx
dx
x 4√ x 2 − 1
∫
dx
x(x 2 + 1)
4. Let a curve C be defined by y = sin −1 (x 2 ), 0 ≤ x ≤ 1. Set up an integral for the length of this curve.
Do not evaluate it.