# Simple Integration

1. Mar 22, 2015

### kishlaysingh

1. The problem statement, all variables and given/known data
integrate

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

3. 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

2. Mar 22, 2015

### axmls

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

3. Mar 22, 2015

### Ray Vickson

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!

4. Mar 24, 2015

### kishlaysingh

ok, fine then....