# Thread: integration of binomial expansion

1. ## integration of binomial expansion

how would you integrate the general binomial expansion??

thanks for and help

2. $\displaystyle \int (a\times x+b)^n \, dx=\frac{(a x+b)^{n+1}}{a(n+1)}$

If $\displaystyle F (x) = \int f(x)dx$ then:
F(ax+b)=F(x)/a

3. ## thanks

thanks but i was more asking how you would integrate the right hand side of the equation? sorry

4. Originally Posted by hmmmm
thanks but i was more asking how you would integrate the right hand side of the equation? sorry
Since the sum is finite you may interchange the summation and the integral so $\displaystyle \int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^k=\sum_{k=0}^{n}\int\left\{{n\choose{k}}x^{n-k}y^k\right\}$

5. ## thanks

so how do we integrate this?

6. $\displaystyle \sum_{k=0}^{n} {n\choose{k}}x^{n-k}y^k$

$\displaystyle = x^n+nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +\frac{n(n-1)\cdots(n-r+1)}{r!}x{n-r}y^r+\cdots+y^n$

So taking:

$\displaystyle \int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^kdx$

Is just a case of using the addition and power rules for integrals on this series.