evaluated, though, using when() to implement the absolute value function:
x^2-1→f(x)
á(when(f(x)≥0,f(x),-f(x)),x,-2,3)
which eventually returns the approximate answer of 9.3333 3370 4891 4. The exact answer of 28/3
can be found by integrating x^2 - 1 over the three intervals [-2,-1], [-1,1] and [1,3], and summing the
absolute values of the results. So, the approximate answer is good to about seven significant digits.
This method works only in Approx mode. In Exact mode, the method still returns the original integrand.
A more accurate Approx mode result can be obtained without using this method, by integrating over
the three intervals listed above. In this case, the result is correct to all 14 significant digits.
(Credit to TM)
[2.4] Use Exact mode to ensure Limit() success
This is mentioned in the manual, but it is worth repeating here, because you can get wrong results
without warning. For example, consider this limit, expressed in two different ways:
limit((1+.05*x/n)^n,n,∞)
limit((1+x/(20*n))^n,n,∞)
In Exact mode, both of these limits result in e^(x/20), which is correct. In Approx or Auto mode, the
second example works, but the first example returns zero.
(Credit to Bhuvanesh Bhatt)
[2.5] Try manual simplification to improve symbolic integration
The CAS may fail to integrate for some expressions, but can succeed if the integrand is expressed
differently. For example, the CAS cannot find a solution to this integral:
á(√(1/(r^2-x^2)),x)
but the correct result is quickly returned for this version:
á(1/√(r^2-x^2),x)
(Credit to Bhuvanesh Bhatt)
[2.6] CAS uses common convention for integral of x^n
In general,
¶ x
n
dx =
x
n+1
n+1
is true, except for n = -1, where the result is undefined. Actually,
2 - 3
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