# Integral/summation proof via induction

• Feb 5th 2010, 08:45 AM
Runty
Integral/summation proof via induction
Suppose $\displaystyle P$ is a polynomial of degree $\displaystyle k\geq 1$. Prove that

$\displaystyle \int P(x)e^x dx = e^x\sum_{j=0}^{k}(-1)^jP^{(j)}(x)+C$

by induction on $\displaystyle k$. Here, $\displaystyle P^{(j)}$ is the $\displaystyle jth$ derivative of $\displaystyle P$, and $\displaystyle P^{(0)}$ is to be interpreted as $\displaystyle P$.

A hint I was provided was to use integration by parts.
• Feb 5th 2010, 09:09 AM
felper
I'll let you the case $\displaystyle k=1$. Now, we'll suposse that the proposition is true for some poyinomial of degree $\displaystyle k$. We have to prove that this proposition is true for $\displaystyle k+1$. Let be $\displaystyle \underbrace{P(x)}_{k+1}$ a polynomial of degree $\displaystyle k+1$.

$\displaystyle \int\underbrace{P(x)}_{k+1}e^xdx$

Integrating by parts

$\displaystyle \int\underbrace{P(x)}_{k+1}(e^x)'dx=\underbrace{P( x)}_{k+1}e^x-\int\underbrace{P(x)}_{k}e^xdx$

But the proposition is true for the polynomial of degree k, then:

$\displaystyle \int\underbrace{P(x)}_{k+1}e^xdx = \underbrace{P(x)}_{k+1}e^x- e^x\sum_{j=0}^k(-1)^j\underbrace{P^{(j)}(x)}_{k}$

And now, conclude (Wink)