1 / 7

COMPUTING ANTI-DERIVATIVES (Integration by PARTS )

COMPUTING ANTI-DERIVATIVES (Integration by PARTS ). The computation of anti-derivatives is just an in- tellectual challenge, we know how to take deriv-atives , but … can we invert the process? We call this Computing the indefinite integral .

trixie
Download Presentation

COMPUTING ANTI-DERIVATIVES (Integration by PARTS )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. COMPUTINGANTI-DERIVATIVES(Integration by PARTS ) The computation of anti-derivatives is just an in-tellectual challenge, we know how to takederiv-atives, but … can we invert the process? We call this Computing the indefinite integral . In the last presentation we have seen a few indefinite integrals (we called them bricks), but they did not include the anti-derivative of many functions! We are going to try and do better !

  2. It pays off to look at differentiation and integration as inverse processes, that is, if we apply each in order we end up (essentially) where we started. First D then I gives us “where we started + C”

  3. First I then D gives us “where we started.” This very simple observation is going to give us a fair amount of power, because we know how D works, and we can essentially take advantage of “undoing” it ! Here we go.

  4. We know how D works on products (the product rule.) It says Apply to both sides (remember that where you started (never mind C !). We get Since, luckily, the process is additive, we get (replacing the symbol with ) which we rewrite as

  5. Now what? Well, in several appropriate circum-stances, if one chooses judiciously, the indef-inite integral one gets on the right-hand side is easier to compute than the left-hand side! Here is an example: you have to compute Choose whence

  6. We get Therefore (check it out, take the derivative !) If you choose the “unwise” , you end up with a worse integral than you started. Try Unfortunately I cannot tell you what appropriate circumstances means, nor what is the judicious choice of , you just have to learn by experi-ence! There is a mnemonic help I can share:

  7. Once I have decided (rightly or wrongly !) what Is, I make myself this little figure and write Your textbook delays this method to much later, p. 488, because it is most useful for integrating transcendental (exponents, logarithms) functions.

More Related