{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Techniques in Integration - Contents 11 Techniques of...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Contents 11 Techniques of Integration 168 11.1 Basic Substitution and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 11.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.3 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
CONTENTS 2
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}