• Notes
• 121

This preview shows page 36 - 40 out of 121 pages.

When we decide which part to differentiate, we need to prioritize. Here is amnemonic to help you remember which parts are the best to pick: LATE.1. Logarithmic (or inverse trig functions)2. Algebraic3. Trigonometric4. ExponentialThis is in order of priorities. In general, if one part has a known derivativebut unknown antiderivative, that’s the one you want to differentiate.Example 1Always start by simplifying the integrand. Referring to ourmnemonic, we differentiate the lnxand integrate thex:Ze1lnxxdx=Ze1xlnx dx=x22lnx|e1-Ze1x221xdx=e22-x24e1=e2+ 14.
2.8.PARTIAL FRACTIONS372.7.4Cyclic DerivativesSometimes, if we differentiate a function a few times, we end up back witha multiple of the original function. The silliest example is (ex)0=ex, but wealso have and (sinx)00= (cosx)0=-sinxamong a few others. This phe-nomenon provides a way to compute integrals involving products of functionslike this.ExampleZπ/20e-xcosx dx=e-xsinx|π/20-Zπ/20-e-xsinx dx=e-xsinx|π/20--e-x(-cosx)|π/20-Zπ/20e-x(-cosx)dx!Zπ/20e-xcosx dx+Zπ/20e-xcosx dx=e-xsinx|π/20-e-xcosx|π/202Zπ/20e-xcosx dx=e-π/2-(-1)Zπ/20e-xcosx dx=e-π/2+ 12.The method is to repeat integration by parts until you get a multiple of theoriginal integral, and then you combine terms to solve.2.8Partial FractionsAnd now for a wonderfully simple method that is usually startling straight-forward. It is easy to make examples where the application of this methodgets confusing, but I don’t see much point in torturing you with those.2.8.1Simple AlgebraBehold:1x(x-1)=1x-1-1xTherefore,Z1x(x-1)dx=Z1x-1dx-Z1xdx= ln(x-1)-lnx+C.
38CHAPTER 2.INTEGRATIONIn general, if you can factor the denominator, then you can split it into asum like this. Here’s how it works when there are two linear factors.ax+b(x-r)(x-s)=Ax-r+Bx-s.Simply clear the denominatorsax+b=A(x-s) +B(x-r)and plug in firstx=rthenx=sto cancel outBthenAfor an easy solution:ar+b=A(r-s)A=ar+br-sas+b=B(s-r)B=as+bs-rThis method of plugging in convenient values ofxis sometimes called the“cover-up” method (since you can cover up a term with your finger whiledoing it on paper) and is also named after Oliver Heaviside. Wouldn’t it benice to get your name on something so obvious yet cunning?Remark:Before going into more elaborate uses of this method, recallthat at the beginning of the course we discussed telescoping sums. This isanother way to use partial fractions:NXk=11k2+k=NXk=11k-1k+ 1= 1-1N+ 1,which gives us the lovely factXk=11k2+k= 1.If the degree of the numerator is greater than or equal to that of thedenominator, then you should first divide.I think the simplest way to do
2.9.GEOMETRIC APPLICATION OF INTEGRALS39this is to add zero and simplify. For example,x3+xx2-x-6=x3-x2-6x+ (x2+ 6x) +xx2-x-6=x(x2-x-6) +x2+ 7xx2-x-6=x+x2+ 7x-8x-6 + (8x+ 6)x2-x-6=x+x2-x-6 + (8x+ 6)x2-x-6=x+ 1 +8x+ 6x2-x-6.If you are more comfortable with synthetic division or polynomial long divi-sion, please go for it! If you’d like to learn one of those approaches instead,
• • • 