Spring 2012
MATH 1220-002: Extra Credit
Name:
Robbie Snellman
Instructions: Students are allowed to work with other students in the class, however, the
tutoring center may not be used for assistance on this assignment. Any student caught using
the tutoring center will forfeit their extra credit. Students may use textbooks or the internet
to research certain topics.
Problem 1: Let
∞
X
fn xn be a power series with fn the Fibonacci numbers, i.e. f0 =
n=0
1, f1 = 1, and fn+2 = fn+1 + fn . Show that the following formula holds:
∞
X
n=0
fn xn =
1
.
1 − x − x2
(Hint: Try multiplying 1 − x − x2 into the series).
1
The following problems are intended to teach you an integration technique known as
tabular integration. This method allows one to integrate by parts very quickly, assuming
the integrand is of a special form.
1
Tabular Integration
Suppose we have an integral of the following form:
Z
f (x)g(x)dx
where we know that f (n) (x) = 0 for some positive integer n, and we can integrate g(x) n + 1times. Tabular integration tells us how to compute our desired integral without using any u’s
or dv’s. Let f (x), f ′(x), f ′′ (x), f (3) (x), . . . , f (n−1) (x) denote the function f and its nonzero
derivatives. Furthermore, let g(x), G1 (x), G2 (x), . . . , Gn+1 (x) denote the function g and its
(n + 1)-st antiderivatives. The following table will allow you to read-off the antiderivative of
f (x)g(x) immediately:
f (x) −→ G1 (x)
f ′ (x) −→ G2 (x)
f ′′ (x) −→ G3 (x)
f (3) (x) −→ G4 (x)
..
.
f (n−1) (x) −→ Gn (x)
f (n) (x) = 0 −→ Gn+1 (x).
Then, the technique of tabular integration gives
Z
f (x)g(x)dx = f (x)G1 (x) − f ′ (x)G2 (x) + f ′′ (x)G3 (x) − . . . + (−1)n−1 f (n−1) (x)Gn (x) + C.
Example. Suppose we want to compute
Z
x2 ex dx.
We know that we should integrate by parts twice to obtain the solution, however, we now
know that we can apply tabular integration with f (x) = x2 and g(x) = ex . Thus, by tabular
2
integration we have
Z
x2 ex dx = x2 ex − 2xex + 2ex + C.
The following problems are intended to familiarize you with the technique of tabular
integration as well as providing you some extra credit.
3
Problem 2: Use tabular integration to find the following antiderivative:
Z
x3 e−x dx
Problem 3: Use tabular integration to find the following antiderivative:
Z
x5 cos(x)dx
4
Problem 4: Explain why tabular integration will not work to find the following antiderivative:
Z
ex sin(x)dx
Problem 5: Use tabular integration to find the following antiderivative:
Z
x4 sin(x)dx
5