# Thread: How to integrate this function?

1. ## How to integrate this function?

Hello

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

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 $\displaystyle x^(550) * (1-x)^(450)$.

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 $\displaystyle (1-x)^{450}= \sum_{i=0}^{450}C(450 i) (-1)^{450-i}x^i$, where C(n, i) is the binomial coefficient, which is the same as $\displaystyle \sum_{i=0}^{450} C(450,i) (-1)^ix^i$ since 450-i is odd if and only if i is.

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

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

$\displaystyle \int{\sum_{n=0}^{450}}{_{450}}C_n(-1)^nx^{n+450}dx$
$\displaystyle =\sum_{n=0}^{450}\int{_{450}C_n}(-1)^nx^{n+550}dx$
$\displaystyle =\sum_{n=0}^{450})\frac{_{450}C_n(-1)^nx^{n+551}}{551}+C$