Integration Techniques
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Integration Techniques
No silver bullet. Here is the review of basic integration techniques, together with some of the challenging problems presented at University of North Texas Integration Bee. You will see that even computer algebra systems such as Maple and Mathematica are not tough enough to complete them. Nevertheless, you could outperform smart software with the basic techniques we have learned.
Derivatives and integrals of basic functions.
Functions | Derivatives | Indefinite integrals |
Fundamental theorem
of calculus |
||
Power and logarithmic
functions |
||
() | ||
Exponential functions | ||
Trigonometric functions | ||
Hyperbolic functions | ||
Inverse trigonometric
functions |
||
Inverse hyperbolic
functions |
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Formulas and identities of basic functions.
Logarithmic and
Exponential functions |
||
Trigonometric functions | ||
where and . | ||
Hyperbolic functions | ||
You may find them useful in the following ``half-angle tangent''
substitution:
Substitution | Identities | ||
, | ; | , , | . |
Strategy for integration with sample problems.
A. Substitution rules.
Substitution | Identities | ||
, | ; | , | . |
; | , | . | |
; | , | . |
B. Integration by parts:
C. Integration of rational functions.
; | . |
; | . |