Integration Homework: Substitution, Partial Fraction, By Parts

[kishlaysingh]

Homework Statement

integrate

Homework Equations

please solve this using methods only like
1. Substitution
2.Partial fraction
3.By Parts

The Attempt at a Solution

i have tried all the above three methods mainly using substitution and by parts...
i have expanded the a^3 - x^3 and then kept the a/2 - x as "p" but still the function doesn't simplify

 

[axmls]

What context did this appear in? It doesn't appear to have an elementary antiderivative.

 

[Ray Vickson]

Homework Statement

integrate
View attachment 80770

Homework Equations

please solve this using methods only like
1. Substitution
2.Partial fraction
3.By Parts

The Attempt at a Solution

i have tried all the above three methods mainly using substitution and by parts...
i have expanded the a^3 - x^3 and then kept the a/2 - x as "p" but still the function doesn't simplify

The integral is "non-elementary", so cannot be done in terms of a finite expression involving only elementary functions (such as powers, roots, trigonometric functions and their inverses, exponentials and logarithms, etc.). However, it can be expressed in terms of the non-elementary "elliptic functions", or it can be expressed in terms of an infinite series of elementary functions.

Note: the non-elementary nature of the integral is a theorem that has been proven rigorously. It is not just a matter of somebody not being smart enough to figure out how to do the integral (in finitely many terms and factors of elementary functions); rather, it has been proven to be impossible. So, even if you allow yourself to use a formula taking 10 billion pages to write out in detail, it still would not be the answer to your question!

 

[kishlaysingh]

ok, fine then...