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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 .

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COMPUTING ANTI-DERIVATIVES (Integration by PARTS )

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  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.

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