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[?] Use substitution (or other) method to get a specific form integral?

Roberto Rivero
Posted 6 years ago

Hi, from a book need to solve an integral: enter image description here , was handwrite solved, but Mathematica show the answer in different form, with code:

Integrate[x^3 (2 - x^2)^12, x]

and

Integrate[x^3 (2 - x^2)^m, x] /. m-> 12

get this: mathematica-show

but the question is, can get a form that looks like this?:

desire form

which is the answer by handwrite and solution book. Thanks!
POSTED BY: Roberto Rivero
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This is sightly different from the method provided by Hans Dolhaine, and slightly different from the desired form, but it might suffice depending on requirements.

jj = Integrate[x^3 (2 - x^2)^12, x];
divisor = x^2 - 2;
pq = PolynomialQuotientRemainder[jj, divisor, x];
Factor[pq[[1]]]*divisor + pq[[2]]

(* Out[3011]= 4096/91 + 1/364 (-2 + x^2)^13 (2 + 13 x^2) *)
POSTED BY: Daniel Lichtblau

Yep. You can get that, but it is somewhat, say, artificial

jj = Integrate[x^3 (2 - x^2)^m, x] /. m -> 12


(jj /. (2 - x^2) -> p /. (2 + 13 x^2) -> q /. 
q -> ((2 - x^2) (PolynomialQuotient[2 + 13 x^2, 2 - x^2, x] + 
PolynomialRemainder[2 + 13 x^2, 2 - x^2, 
x]/(2 - x^2))) /. (2 - x^2) -> p // Expand) /. 
p -> 2 - x^2
POSTED BY: Hans Dolhaine
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