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Techniques in Integration

# Techniques in Integration - Contents 11 Techniques of...

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Contents 11 Techniques of Integration 168 11.1 Basic Substitution and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 11.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.3 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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CONTENTS 2
Chapter 11 Techniques of Integration 11.1 Basic Substitution and Formulas Definition The indefinite integral of a function f , written Z f ( x ) dx, is the most general antiderivative of f ; that is, Z f ( x ) dx = F ( x ) + C ( C is any constant) if and only if F 0 ( x ) = f ( x ). Basic Formulas: (Memorize this list) Z u n du = u n +1 n + 1 + C ( n 6 = - 1) Z du u = ln | u | + C, if u 6 = 0 Z e u du = e u + C Z a u du = a u ln a + C Z sin u du = - cos u + C Z cos u du = sin u + C Z sec 2 u du = tan u + C Z csc 2 u du = - cot u + C Z sec u tan u du = sec u + C Z csc u cot u du = - csc u + C Z du 1 - u 2 = sin - 1 u + C Z du 1 + u 2 = tan - 1 u + C

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169 11.1 Basic Substitution and Formulas Z du | u | u 2 - 1 = sec - 1 u + C, | u | > 1 Z tan u du = - ln | cos u | + C or ln | sec u | + C Z cot u du = ln | sin u | + C Z sec u du = ln | sec u + tan u | + C Z csc u du = ln | csc u - cot u | + C Z [ f ( u ) + g ( u ) du ] = Z f ( u ) du + Z g ( u ) du Z kf ( u ) du = k Z f ( u ) du if k is a constant 1. Z ( x 2 + a ) 2 dx 2. Z ( x - 3) 2 dx 3. Z x 5 2 dx 4. Z aπ dx x 5. Z [( x + 1) 3 + e x ] dx 6. Z x 2 e x + x x 2 dx 7. Z 10 sec 2 x dx 8. Z 16(2 x + 1) dx 9. Z (sin x + cos x ) dx 10. Z ( x + sec x tan x ) dx 11. Z (sec 2 x + 1) dx 12. Z x 2 + 3 x 2 3 2 + 1 x 2 ! dx 13. Z 5 x - 4 x 3 + 3 x 5 dx 14. Z x dx 15. Z x x dx 16. Z x x - 1 x dx 17. Z x x dx 18. Z x 5 + x 3 + 2 1 + x 2 dx ( Hint: divide) 19. Z x 3 - a 3 x - a dx x 3 3 + a 2 x 2 + a 2 + x + A 20. Z ( x 3 + 27) dx x 2 - 3 x + 9 21. Z 1 - x 2 1 - x 4 dx tan - 1 x + c 22. Z x + 1 1 - x 2 dx 23. Z x 1 - x 2 + 4 1 - x 2 ! dx 24. Z x 3 - 8 x 2 + 2 x + 4 dx 25. Z x 3 + x 2 - x - 1 x - 1 dx 26.
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