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.
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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.
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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
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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
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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
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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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