# Thread: How to integrate this function?

1. ## How to integrate this function?

Hello

I'm trying to find the indefinite integral of $x^{550} * (1-x)^{450}$.

How do I do this? I tried by parts but it seems to be about 450 iterations long! Any good substitution tricks or am I missing something?

Thanks!

Thomas

2. Originally Posted by Tornam
Hello

I'm trying to find the indefinite integral of $x^(550) * (1-x)^(450)$.

How do I do this? I tried by parts but it seems to be about 450 iterations long! Any good substitution tricks or am I missing something?

Thanks!

Thomas
No substitutions. You could try using the binomial theorem to say that $(1-x)^{450}= \sum_{i=0}^{450}C(450 i) (-1)^{450-i}x^i$, where C(n, i) is the binomial coefficient, which is the same as $\sum_{i=0}^{450} C(450,i) (-1)^ix^i$ since 450-i is odd if and only if i is.

Then $x^{550}(1- x)^{450}= \sum_{i=0}^{450} C(450,i) (-1)^i x^{i+ 550}$ and integrate term by term.

3. Ah yes. What do you mean term by term though? As in integrate every term in the sum?

Thanks for your help!!

4. Originally Posted by Tornam
Ah yes. What do you mean term by term though? As in integrate every term in the sum?

Thanks for your help!!
$\int{\sum_{n=0}^{450}}{_{450}}C_n(-1)^nx^{n+450}dx$

$=\sum_{n=0}^{450}\int{_{450}C_n}(-1)^nx^{n+550}dx$

$=\sum_{n=0}^{450})\frac{_{450}C_n(-1)^nx^{n+551}}{551}+C$