Some tough integrals...?

Question

I have two integrals from my calculus homework that I can't figure out how to do and I don't have any software that does symbolic calculus. If anyone could try these on Mathematica or Maple, that would be awesome. Here they are

(1-y^2)sin(y^3)

and

(1-x^2)sqrt(x^3+1)

Answer 1

No wonder you had troubles with these! Both are non-
elementary. Let's do each by splitting it in two.
The first one gives
int[sin(y^3) dy] - int[y^2 sin(y^3) dy]
The second integral is easy to do by substitution.
The first one, however is non-elementary.
integrals.wolfram.com evaluates it in terms of
the incomplete gamma function.
Do the second one the same way.
You get
int[ sqrt(x^3+1) dx] - int[x^2 sqrt(x^3+1) dx].
Again the second integral is easy to do by
substitution.
But the first one is the integral of the
square root of a cubic polynomial, which
makes it an elliptic integral and nonelementary.

Answer 2

These are tough, even for Maple.

Send me a message and I'll send you the results in a text file. It won't format properly in here.

I should point out, Maple used special functions to describe both integrals.

Answer 3

How about solving a simpler problem first?
Try

y^2sin(y^3)

Doesn't this look more palatable? The idea is to solve a
simpler related problem, as described by Polya in the
reference.