This integral made my Ti-Nspire CAS freezeee

[T Tran]

integrate ( (sin(x^2) - x^2 * cos(x^2)) / x^4 ,x , 0.0 , infinity)

I put 0.0 to make the calc do a numeric evaluation.
But the calc then froze up.

[T Tran]

expected answer should be
(1/3)*(Pi/2)^(1/2)

[Wayne]

I imagine that you want to make a point, but I have no idea what it
is. I gave this integral to Maple 13 and even it did not evaluate the
integral to the value that you suggest is correct.

[T Tran]

I've never seen the calculator froze up. Usually, it would return the
input whenever it cannot solve.
Mathematica can do that problem though

http://www.wolframalpha.com/input/?i=integrate+(+(sin(x^2)+-+x^2+*+cos(x^2))+/+x^4+,x+)

http://www.wolframalpha.com/input/?i=integrate+(+(sin(x^2)+-+x^2+*+cos(x^2))+/+x^4+,x+,0,+infinity)

[Wayne]

Tran,

There are going to be many integration problems that your calculator
cannot solve. This one is particularly difficult. The only way I
have been able to verify your result analytically without the use of a
calculator is to notice that the integrand of the improper integral
that you present is related to the derivative of the Bessel function
of the first kind of order 3/2. That, together with some quite
involved manipulation and the knowledge that the Fresnel integral
converges, allowed me to demonstrate analytically that the improper
integral in question converges to sqrt(pi)/(3sqrt(2)).

The actual result is not nearly as important as that you understand
that the Nspire is going to often be "stumped" by such complex
problems. The point is that our students are unlikely to be able to
understand or to appreciate the solution of such problems until they
have at least studied the theory of special functions and some other
related material. This material is not even normally studied by
graduate students in mathematics unless they intend to specialize in
the field. Few do that these days. Those that do will have
substantially more powerful tools than the Nspire available to them
when they are needed. Mathematica and Maple are two examples.

For our students, normally in high school and undergraduate colleges
and universities, the Nspire is almost always adequate mathematically
and it also has the distinct advantage of having very good teaching
tools. I definitely want TI to continue to add mathematical
capability to the device using the code from Derive, but our students
will be fine in any case. TI now has a very good base on which to add
all the mathematical capabililty that was available in Derive,
including the many special functions like Bessel. Let's hope they
continue to invest in the technology and continue to provide us with
good mathematics as well as good teaching tools.

Wayne

[Nelson Sousa]

when you're trying to compute a numerical integral use nint instead of the integral template. The template will try symbolic manipulations and even with approximate bounds, like you did, only the final evaluation will be approximate.

using nint( function,x,0,10) I got 0.418027 in about 5 seconds. It's not very precise, but nevertheless it's within 0.01 of the expected result.

Beware that as this function is oscilating violently you must keep the integration interval small (nint will decrease the step when the function is smoother and decrease it when it oscilates)

Nelson

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