# Thread: Help with reduction formula

1. ## Help with reduction formula

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)

2. Originally Posted by player07
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)

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 $\displaystyle \frac{x}{11} = \cosh t$ ....

3. Hi mr fantastic, maybe you have missunderstand what i have written
i will rewrite the question
the reduction formula for this :

can you look again to the question please.

4. He really didnt misunderstnad you.
You have misunderstood him!

5. thank you for you help