Download Workshop #4, Math 162 (Spring 2017) Week of February 13th, 2017

Workshop #4, Math 162 (Spring 2017)
Week of February 13th, 2017.
Today’s topics:
• Integration by parts
• Trigonometric substitution
• Trigonometric integrals
• Partial fractions
Warm-up question Assume that the density of the water is 1000 kg/m3 and the acceleration due to gravity is g m/s2 .
A tank in the shape of an inverted cone has a height of 20 meters and a base radius of 5 meters and is filled with water
to a depth of 15 meters.
(a) Draw the cylinder in a coordinate system, call the variable x, that runs from 0 at the top of the tank to 20 at the
bottom, and compute all relevant quantities (radius/volume/mass of a typical slice, force/work required to move the
slice to the top). Then set up an integral in that coordinate system that would determine the amount of work needed
to pump all of the water to the top of the tank.
(b) Now redo the problem with a variable, call it y, that starts at 0 at the base of the cone and increases to 20 at the top
of the cone.
(c) Do the two integrals you set up have different or the same values? (Hint: you do not need to evaluate them to answer
this question.)
Workshop questions
1. What are some ways in which you will prepare for the exam next week? When will you start?
2. Come up with a list of steps you might take when confronted with an integral (definite or indefinite). What kind of
things would you check for? What methods of integration have you seen? Which methods would you try to use first?
Do you prefer to do the indefinite integral and then evaluate or carry the endpoints through? Try to use your steps
when confronting the following integrals. (If you are sure you know how to do one, you do not need to carry it out
completely; if unsure, do work it out completely.) Try to group together integrals that require similar approaches:
ˆ
ˆ
(a)
x arctan(x)dx
(l)
sec4 (x) tan4 (x)dx
ˆ
ˆ 2
(m)
sin4 (x) cos4 (x)dx
(b)
x ln(x)dx
ˆ1
ˆ
√
x x + 1 dx
(c)
ˆ
x2
(d)
p
ˆ
(e)
ˆ
(x2
x2 + 1 dx
1
dx
+ 36)2
x5
3 dx
(36x2 + 1) 2
2
ˆ ln(x)
(g)
dx
x
ˆ
(h)
e6x sin(3x)dx
ˆ
(i)
e6x sin(e3x )dx
ˆ
(j)
sec(x) tan2 (x)dx
ˆ
(k)
sec10 (x) tan3 (x)dx
(f)
π
2
x sin(x) cos(x)dx
(n)
ˆ
π
4
x
dx
25x2 − 4
ˆ p
(p)
x 25x2 − 4dx
ˆ √
25x2 − 4
dx
(q)
x
ˆ
1
p
(r)
dx
1 + (6x − 7)2
ˆ
−7x2
√
(s)
dx
4x − x2
ˆ
p
(t)
e4x 1 + e2x dx
√
(o)
ˆ
− 52
(u)
− 45
√
25x2 − 4
dx
x
(v) Make up an integral for your partner/group to do
3. In this problem you will practice some of the algebra behind the method of partial fractions. If you have not yet
seen this method of integration in class, do not worry. This workshop problem involves mostly algrebra.
(a) Factor x2 + x + 1, x2 + 4x − 21 , x2 + 6, and x2 − 25 into linear factors if possible. How do you tell if a quadratic
polynomial will factor or not? What do you call it if it will not factor?
(b) Factor x3 + 3x2 − 25x + 21, x3 − x2 + 6x − 6, and x3 − 1 completely using that x = 1 is a root of each. Will
a cubic polynomial always factor into linear factors? Will a cubic polynomial always have at least one linear
factor? (Challenge: Can you prove it?)
(c) Factor x6 + 2x5 − 3x4 − 4x3 + 4x2 into linear factors, using the fact that x = 1 is a root of multiplicity two.
(d) Solve for c and d in the following fractional equation:
c
d
6x
=
+
x2 − 25
x+5 x−5
What steps did you use?
(e) Perform polynomial long division:
x2 − 25 ) 3x3 + 3x2 − 69x − 75
(f) Use (e) to write
3x3 + 3x2 − 69x − 75
as an “integral part” ax + b and a remainder p(x) divided by x2 − 25
x2 − 25
3x3 + 3x2 − 69x − 75
p(x)
= ax + b + 2
2
x − 25
x − 25
(g) Use (d) to further expand this into
c
d
3x3 + 3x2 − 69x − 75
= ax + b +
+
x2 − 25
x+5 x−5
(h) Now find the following integral if you can
ˆ
3x3 + 3x2 − 69x − 75
dx
x2 − 25
What were all the steps you used above?
Review Problems Find each of the following integrals using as many methods as possible.
ˆ
ˆ
(a)
sec(x)dx
(f)
arctan(x)dx
ˆ
ˆ
sec2 (x)dx
(b)
(g)
ˆ
sec3 (x)dx
(c)
ˆ
ˆ
(h)
p
x x2 + 9 dx
sec(x) tan(x)dx
(d)
ˆ
ˆ
(e)
p
x x2 + 1 dx
tan(x)dx
(i)
sin3 (x) cos4 (x)dx