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Thread: Integral/summation proof via induction

  1. #1
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    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.
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  2. #2
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    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
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