How do I obtain a set of orthogonal polynomials up to the 7th term?

Barracuda
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Hello everyone,
I need some help with this solution. I'm trying to obtain a set of orthogonal polynomials up to the 7th term. I think i got it up to the 6th term, but the integration is getting more complex. I'm not sure if I'm on the right track. Please help
 
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Could you write the recursion relation, preferably in Latex?
 
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Nobody wants to wade through your hand-written figures. If you want a response, you're going to have to learn how to enter it into latex.
 
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Infrared said:
I haven't read your images, but it seems to me like you're describing the Legendre polynomials (https://en.wikipedia.org/wiki/Legen...ition_by_construction_as_an_orthogonal_system). If so, you can check your work there.

Is there a reason you have to do this calculation by hand (instead of using a computer)? I would expect it to be very tedious.
I'm using Gram-Schmidt. I'll type it out using Latex
 
If you mean typing out all your handwriting, first please explain why you seem to be doing it the hard way (rather than using the recurrence relation for example).
 
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