Tough integration (1 Viewer)

[rama_v]

Here's a hard integration our calculus lecturer gave us the other day:
∫x⁴(1-x)⁴ / (1 +x²) dx, from 0 to 1. Can anyone show me a way of doing the integral by hand which is not pages and pages long? btw the answer is 22/7 - pi.

 

[Riviet]

I've done a similar question before, but my knowledge of calculus only allows me to do it the long way. It turns out the long way of expanding and long division works out alright, the integration part is also fairly straight forward.

 

[rama_v]

I've done a similar question before, but my knowledge of calculus only allows me to do it the long way. It turns out the long way of expanding and long division works out alright, the integration part is also fairly straight forward.

Yeah. I can do it the long way. And it is long lol. Just wondering if there is any other, neater solution.

 

[KeypadSDM]

Here's a hard integration our calculus lecturer gave us the other day:
∫x⁴(1-x)⁴ / (1 +x²) dx, from 0 to 1. Can anyone show me a way of doing the integral by hand which is not pages and pages long? btw the answer is 22/7 - pi.

There's a difference between doing it the short way and KNOWING there's a short way.

Do you know there's a short way?

 

[rama_v]

There's a difference between doing it the short way and KNOWING there's a short way.

Do you know there's a short way?

Sorry, I should have made it clear. I don't know of whether there is a short way.

In any case, I think we agree now that polynomial division is probably the easiest albeit long method.

 

[Mill]

That actually leads to a 'cute' result.

You can easily see that the integral must be positive.

Thus, 22/7 - pi > 0

22/7 > pi

pi != 22/7



It's cute in the sense that 22/7 is a 'well-known' approximation for pi.

 

[OmegaSTealth]

I tried this one by long division, didn't end up using more than 18 lines (including long division working, so about 5 or 6 lines of integration working) - that is not what i would call a long solution that is "pages long" - 1/2 pg at most.
you end up with a seventh degree polynomial with a 4arctan(x) at the end.

 

[withoutaface]

int(x^4(1-x)^4/(1+x^2))dx = int((x^8-4x^7+6x^6-4x^5+x^4)/(1+x^2)
then break it up = int ((x^8+x^6)/(1+x^2) - (4x^7+4x^5)/(1+x^2)+(5x^6+5x^4)/(1+x^2) -(4x^4+4x^2)/(1+x^2) +(4x^2+4)/(1+x^2) - 4/1+x^2)dx
= int(x^6-4x^5+5x^4-4x^2+4 - 4/(1+x^2))dx
and then it falls out. Hardly 5 pages :/