Chap7_Sec6

# Chap7_Sec6 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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TECHNIQUES OF INTEGRATION 7

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TECHNIQUES OF INTEGRATION 7.6 Integration Using Tables and Computer Algebra Systems In this section, we will learn: How to use tables and computer algebra systems in integrating functions that have elementary antiderivatives.
However, you should bear in mind that even the most powerful computer algebra systems (CAS) can’t find explicit formulas for: The antiderivatives of functions like e x 2 The other functions at the end of Section 7.5

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TABLES OF INTEGRALS Tables of indefinite integrals are very useful when: We are confronted by an integral that is difficult to evaluate by hand. We don’t have access to a CAS.
TABLES OF INTEGRALS A relatively brief table of 120 integrals, categorized by form, is provided on the Reference Pages.

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TABLES OF INTEGRALS More extensive tables are available in: CRC Standard Mathematical Tables and Formulae, 31st ed. by Daniel Zwillinger (Boca Raton, FL: CRC Press, 2002), which has 709 entries Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products, 6e (San Diego: Academic Press, 2000), which contains hundreds of pages of integrals
TABLES OF INTEGRALS Remember, integrals do not often occur in exactly the form listed in a table. Usually, we need to use substitution or algebraic manipulation to transform a given integral into one of the forms in the table.

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TABLES OF INTEGRALS The region bounded by the curves y = arctan x , y = 0, and x = 1 is rotated about the y -axis. Find the volume of the resulting solid. Example 1
TABLES OF INTEGRALS Using the method of cylindrical shells, we see that the volume is: 1 0 2 arctan V x x dx π = Example 1

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TABLES OF INTEGRALS In the section of the Table of Integrals titled Inverse Trigonometric Forms, we locate Formula 92: 2 1 1 1 tan tan 2 u u u u du u C u - - + = - + Example 1
TABLES OF INTEGRALS So, the volume is: 1 1 0 1 2 1 0 1 2 1 0 1 2 1 2 2 tan 1 2 tan 2 2 ( 1) tan (2 tan 1 1) [2( / 4) 1] V x x dx x x x x x x π - - - - = + = - = + - = - = - = - Example 1

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Chap7_Sec6 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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