Hi, i want to have a reduction formula for this problem i hope that you can help me with this
J[k] = int((x^2-121)^k, x = 0 .. 11)
thanks in advance
Assuming you're familiar with the hyperbolic functions sinh and cosh, I think the simplest approach (and I leave myself wide open to an *ahem* from someone like Krizalid) is to first make the substitution $\displaystyle x = 11 \cosh^2 t$.
This will lead to $\displaystyle J_k = 11^{2k + 1} \int \sinh^{2k+1} t \, dt$ which can be done using a well known reduction formula (that you might find easier to derive):
$\displaystyle \int \sinh^m t \, dt = \frac{\sinh^{m-1}t \, \, \cosh t}{m} - \frac{m-1}{m} \int \sinh^{m-2} t \, dt $.
Then just sub back $\displaystyle \frac{x}{11} = \cosh t$ ....