Integration with computers
Indefinite integration
Polynomials and rational functions
For polynomials, computer algebra systems like Mathematica
do a fine job of antidifferentiating.
Integrate@x ^ 2, xD
x3
3
Note, though, that Mathematica gives only a single antiderivative; it omits the constant of integration.
But as soon as we move beyond polynomials to rational functions, there can be problems.
Integrate@1  x, xD
Log@xD
Note that the computer's answer in this case is only a valid
antiderivative for positive values of x. (FYI, Mathematica uses
Log[x] to mean natural log.)
Let's explore this problem further.
The (natural) log, absolute value, and Heaviside functions
D@Log@xD, xD
1
x
(Well, at least Mathematica is consistent...)
D@Log@Abs@xDD, xD
Abs¢ HxL
x¤
This is true, but not very helpful.
What about antidifferentiating |x|?
2
Integration.nb
What about antidifferentiating |x|?
Integrate@Abs@xD, xD
à x¤ â x
This is Mathematica's way of saying "I don't know how to symbolically antidifferentiate that."
In fact, we know that (1/2) x |x| is an antiderivative of |x|; can we
get Mathematica to at least recognize this?
D@H1  2L x Abs@xD, xD
Abs@xD
1
+
2
x Abs¢ @xD
2
Simplify@%D
1
HAbs@xD + x Abs¢ @xDL
2
Guess not!
Integration by partial fractions
Let's try that integration-by-partial fractions example.
Integrate@H3 x ^ 3 - 3 x ^ 2 + x + 1L  Hx ^ 4 - 2 x ^ 3 + 2 x ^ 2 - 2 x + 1L, xD
1
+ ArcTan@xD + Log@- 1 + xD + LogA1 + x2 E
1-x
D@1  H1 - xL + ArcTan@xD + Log@- 1 + xD + Log@1 + x ^ 2D, xD
1
1
+
2
H1 - xL
1
2x
+
-1 + x
+
1+x
2
1 + x2
We need to tell Mathematica to combine terms:
Together@%D
1 + x - 3 x2 + 3 x3
H- 1 + xL2 I1 + x2 M
Okay, but we had to explicitly prompt Mathematica to write the
answer in a friendly form.
Integration.nb
3
Polynomials revisited
I said before that Mathematica does a fine job of antidifferentiating, but in a sense this isn't true.
D@H1  18L Hx ^ 2 + 5L ^ 9, xD
8
x I5 + x2 M
Integrate@%, xD
390 625 x2
+ 156 250 x4 +
218 750 x6
2
+ 21 875 x8 + 4375 x10 +
3
1750 x12
3
+ 50 x14 +
5 x16
x18
+
2
18
Simplify@% - H1  18L Hx ^ 2 + 5L ^ 9D
1 953 125
18
Here Mathematica has not chosen a good (i.e., nicely factoring)
antiderivative.
Human-computer collaboration on an indefinite integral
Here's an integral that you can do but Mathematica can't.
Integrate@H1 + Log@xDL Sqrt@1 - Hx * Log@xDL ^ 2D, xD
à H1 + Log@xDL
1 - x2 Log@xD2 â x
Here Mathematica fails to guess the correct substitution.
We can help Mathematica out of a jam by giving it the correct
substitution:
Put u = x ln x, du = (1 + ln x) dx.
Integrate@Sqrt@1 - u ^ 2D, uD
1
u
1 - u2 + ArcSin@uD
2
% . u ® x Log@xD
1
ArcSin@x Log@xDD + x Log@xD
2
1 - x2 Log@xD2
4
Integration.nb
D@%, xD
1
x Log@xD I- 2 x Log@xD - 2 x Log@xD2 M
1 + Log@xD
+
2
+
1 - x2 Log@xD2
2
1 - x2 Log@xD2 + Log@xD
1 - x2 Log@xD2
1 - x2 Log@xD2
Simplify@%D
H1 + Log@xDL
1 - x2 Log@xD2
This enhances
answer
1
2
KArcSin@x Log@xDD + x Log@xD
our
confidence
in
the
1 - x2 Log@xD2 O
This is a great example of human and computer working
together to achieve a task that would be hard for a human alone
or a computer alone.
Definite integration
The (natural) log, absolute value, and Heaviside functions
Definite integrals are often handled by different routines in a
computer algebra system than indefinite integrals. For instance,
Mathematica does not always evaluate definite integrals by
using the Evaluation Theorem. We can see this with an
example.
Integrate@1  x, 8x, 1, 2<D
Log@2D
Integrate@1  x, 8x, - 2, - 1<D
- Log@2D
Mathematica gets the right answer for the definite integral, even
though it didn't for the indefinite integral.
(What about integrating from -1 to 1?
Integration.nb
5
Integrate@1  x, 8x, - 1, 1<D
1
Integrate::idiv : Integral of
does not converge on 8-1, 1<. ‡
x
11
à
-1
âx
x
Mathematica correctly reports that the integral doesn't
converge.)
What about integrating the absolute value function?
Integrate@Abs@xD, 8x, - 1, 1<D
1
Ditto.
Can we use "definite" integration with indefinite limits of integration to get an antiderivative by the back door (i.e. via the FTC)?
Integrate@Abs@tD, 8t, 1, x<D
- Hx - 1L HReHxL - 1L
ImHxL2 + HReHxL - 1L2 - ReHxL , IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M -
ImHxL2 , IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M - log
ImHxL2 + HReHxL - 1L2 + ReHxL - 1 +
logI, IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M +
ImHxL2 + ReHxL2 - ReHxLM
32
“ J2 IImHxL2 + HReHxL - 1L2 M N
Ugh!
Simplify@%D
- Hx - 1L HReHxL - 1L
ImHxL2 + HReHxL - 1L2 - ReHxL , IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M -
ImHxL2 , IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M - log
ImHxL2 + HReHxL - 1L2 + ReHxL - 1 +
logI, IImHxL2 I2 ReHxL2 - 2 ReHxL + 1M + ImHxL4 + HReHxL - 1L2 ReHxL2 M +
ImHxL2 + ReHxL2 - ReHxLM
32
“ J2 IImHxL2 + HReHxL - 1L2 M N
To see if this is correct, try to evaluate it at x = 1 and see if we
get (1/2) 1 |1| = 1/2:
6
Integration.nb
% . x ® 1
1
Power::infy : Infinite expression
encountered. ‡
032
¥::indet : Indeterminate expression -¥ + ¥ encountered. ‡
Indeterminate
It appears that Mathematica has created an expression involving
the variable x that it can't evaluate for any particular value of x!
Human-computer collaboration on a definite integral
How does Mathematica do with that u = x ln x example?
Integrate@H1 + Log@xDL Sqrt@1 - Hx * Log@xDL ^ 2D, 8x, 0.9, 1.1<D
1.1
à
HlogHxL + 1L
1 - x2 log2 HxL â x
0.9
It takes Mathematica half a minute or so to find this (useless)
answer.
When we ask Mathematica to do this integral numerically, we
get an answer quickly.
N@%D
0.199331
And this seems to be right, if we apply the Evaluation Theorem
using the antiderivative we found before:
HH1  2L HArcSin@x Log@xDD + x Log@xD Sqrt@1 - x ^ 2 Log@xD ^ 2DL . x ® 1.1L HH1  2L HArcSin@x Log@xDD + x Log@xD Sqrt@1 - x ^ 2 Log@xD ^ 2DL . x ® 0.9L
0.199331
For some ideas about how did Mathematica might have done
the calculation of the definite integral (without knowledge of
the antiderivative), see section 6.5 of Stewart, which discusses
numerical methods of integration.