Symbolic Toolbox: integration

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Igor
Igor 2011-12-12
>> int('cos(x^2)^2')
Warning: Explicit integral could not be found.
ans =
int(cos(x^2)^2, x)
>> int('cos(2*x^2)')
ans =
(pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/2
But there is a formula cos(2y)=2cos(y)^2)-1 and in first case MATLAB can't solve..
May be MuPAD forced to solve?
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Andrei Bobrov
Andrei Bobrov 2011-12-13
I'm agree , Walter!
(Maple Toolbox)
http://imageshack.us/photo/my-images/843/symint3.png/
Andrei Bobrov
Andrei Bobrov 2011-12-13
Last comment :)
http://imageshack.us/photo/my-images/402/symint4.png/
(for Maple Toolbox)

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bym
bym 2011-12-14
Apparently you have to do the substitution manually
int(simple(cos(x^2)^2))
Warning: Explicit integral could not be found.
Using the sincos version returned by
simple(cos(x^2)^2)
yields this
int(cos(2*x^2)/2 + 1/2)
ans =
x/2 + (pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/4
I could not find where the sincos 'simplification' is available outside the simple command as it is in Maple...maybe this is the reason Maple returns the result automagically ;)
[edit for completeness]
int(feval(symengine, 'combine', cos(x^2)^2, 'sincos'))
ans =
x/2 + (pi^(1/2)*fresnelC((2*x)/pi^(1/2)))/4
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Igor
Igor 2011-12-14
Yes, yes...
good construction with "combine"/"sincos"
But from the technical calculation point of view (as semi-analitical and fast integration is needed)
it's not good to think about integration procedure details...
And if more complex integral occures??
To ask own intuition?
Walter Roberson
Walter Roberson 2011-12-14
Humans are better at seeing patterns or understanding context than machines are. It is *often* the case that a human will have to apply knowledge of theoretical equivalences in order to make progress in integration.
MuPAD is not as strong as some of the other products around, but _all_ of them are missing a lot of patterns.

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