Advanced Integration Techniques
Before introducing the more advanced techniques, we will look at a shortcut for the easier of the substitution-type integrals. Advanced integration techniques then follow: integration by parts, trigonometric integrals, trigonometric substitution, and partial fraction decompositions.
Substitution-Type Integration by Inspection
In this section we will consider integrals which we would have done earlier by substitution, but which are simple enough that we can guess the approximate form of the antiderivatives, and then insert any factors needed to correct for discrepancies detected by (mentally) computing the derivative of the approximate form and comparing it to the original integrand. Some general forms will be mentioned as formulas, but the idea is to be able to compute many such integrals without resorting to writing the usual u-substitution steps. Example 7.1.1 Compute cos 5x dx.
Solution: We can anticipate that the approximate form1 of the answer is sin 5x, but then d d sin 5x = cos 5x · (5x) = cos 5x · 5 = 5 cos 5x. dx dx Since we are looking for a function whose derivative is cos 5x, and we found one whose derivative is 5 cos 5x, we see that our candidate antiderivative sin 5x gives a derivative with an extra factor of 5, compared with the desired outcome. Our candidate antiderivative’s derivative is 5 times too large, so this candidate antiderivative, sin 5x must be 5 times too large. To compensate and 1 arrive at a function with the proper derivative, we multiply our candidate sin 5x by 5 . This give us a new candidate antiderivative 1 sin 5x, whose derivative is of course 1 cos 5x · 5 = cos 5x, as 5 5 desired. Thus we have 1 cos 5x dx = sin 5x + C. 5 It may seem that we wrote more in the example above than with the usual u-substitution method, but what we wrote could be performed mentally without resorting to writing the details. In future sections, an integral such as the above may occur as a relatively small step in the execution of a more advanced and more complicated method (perhaps for computing a much more diﬃcult integral). This section’s purpose is to point out how such an integral can be quickly dispatched, to avoid it becoming a needless distraction in the more advanced methods. 1 In
this section, by approximate form we mean a form which is correct except for multiplicative constants.
CHAPTER 7. ADVANCED INTEGRATION TECHNIQUES
The method used in all the examples here can be summarized as follows: 1. Anticipate the form of the antiderivative by an approximate form (correct up to a multiplicative constant). 2. Diﬀerentiate this approximate form and compare to the original integrand function; 3. If Step 1 is correct, i.e., the approximate form’s derivative diﬀers from the original integrand function by a multiplicative constant, insert a compensating, reciprocal multiplicative constant into the approximate form to arrive at the actual antiderivative; 4. For veriﬁcation, diﬀerentiate the answer to see if the original integrand function emerges. For instance, some general formulas which should be quickly veriﬁable by inspection (that is, by reading and mental computation rather than with paper and pencil, for instance) follow: ekx dx cos kx dx sin kx dx sec2 kx dx csc2 kx dx sec kx tan kx dx csc kx cot kx dx 1 dx ax + b 1 kx e + C, k 1 = sin kx + C, k 1 = − cos kx + C, k 1 = tan kx + C, k 1 = − cot kx + C, k 1 = sec kx + C, k 1 = − csc kx + C, k 1 = ln |ax + b| + C. a = (7.1) (7.2) (7.3) (7.4) (7.5) (7.6) (7.7) (7.8)
Example 7.1.2 The following integrals can be computed with u-substitution, but also are computable by inspection: e7x dx = 1 7x e + C, 7 cos x x dx = 2 sin + C, 2 2 1 tan πx + C, π
1 1 dx = ln |5x − 9| + C, 5x − 9 5 1 sin 5x dx = − cos 5x + C, 5
sec2 πx dx =
1 csc 6x cot 6x dx = − csc 6x + C. 6
While it is true that we can call upon the formulas (7.1)–(7.8), the more ﬂexible...
Please join StudyMode to read the full document