Math 121 Assignment 3
Due Friday January 29
Practice problems (do NOT turn in):
• Try out as many problems from Sections 6.1-6.4 as you can, with
special attention to the ones marked as challenging problems.
As a test of your understanding of the material, work out the
problems given in the chapter review. You may skip the ones
that require computer aid.
Problems to turn in:
1. Evaluate the following integrals:
Z
Z
Z
ln(ln x)
2
(a)
dx
(b) (arcsin(x)) dx
(c) xex cos x dx
x
Z
Z
Z
x3 + 1
dt
dx
(d)
dx (e)
(f )
2
2
2
2x
12 + 7x + x
(t − 1)(t − 1)
e − 4ex + 4
Z
Z
Z
dx
dθ
dx
(h)
(i)
(g)
.
3
3
1 + cos θ + sin θ
x2 (x2 − 1) 2
x2 (x2 + 1) 2
2. Use
of undetermined coefficients to evaluate the integral
R 2 the method
x (ln x)4 dx.
3. Write down the form that the partial fraction expansion of
x5 + x3 + 1
(x − 1)(x2 − 1)(x3 − 1)
will take. Do not evaluate the
R constants.
2
4. Consider the integral I = e−x dx. It is known (and you can
use this fact without proof) that if it were possible to evaluate the
integral I using elementary functions, it would take the form
2
I = P (x)e−x + C
where P is a polynomial. Show that such a polynomial P does not
exist. This is called a proof by contradiction, and it shows that
2
an elementary function (such as e−x ) may very well possess nonelementary anti-derivatives.
5. Obtain reduction formulae for
Z π
Z
2
n
In =
x sin x dx
and
Jn = sinn x dx
0
and use them to evaluate I6 and J7 .
1