Results 1 to 2 of 2

Math Help - Integral/summation proof via induction

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    232

    Integral/summation proof via induction

    Suppose P is a polynomial of degree k\geq 1. Prove that

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

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

    A hint I was provided was to use integration by parts.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Feb 2009
    From
    stgo
    Posts
    84
    I'll let you the case k=1. Now, we'll suposse that the proposition is true for some poyinomial of degree k. We have to prove that this proposition is true for k+1. Let be \underbrace{P(x)}_{k+1} a polynomial of degree k+1.

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

    Integrating by parts

    \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:

    \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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. summation Induction
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 23rd 2011, 02:21 AM
  2. Reimann integral proof by induction
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: July 5th 2011, 04:26 AM
  3. Induction proof of summation
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 21st 2010, 09:56 AM
  4. [SOLVED] induction proof of summation
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: November 30th 2009, 02:29 AM
  5. Proof of summation of i^3 by induction, help needed
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 19th 2009, 11:24 PM

Search Tags


/mathhelpforum @mathhelpforum