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SYMMETRY

Sec 5.5. SYMMETRY. THE SUBSTITUTION RULE. Sec 5.5: THE SUBSTITUTION RULE. SYMMETRY. Suppose f is continuous on [-a, a] and even. even. Suppose f is continuous on [-a, a] and odd. Odd. Sec 5.5: THE SUBSTITUTION RULE. Term-102. Sec 5.5: THE SUBSTITUTION RULE. Term-091.

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SYMMETRY

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  1. Sec 5.5 SYMMETRY THE SUBSTITUTION RULE

  2. Sec 5.5: THE SUBSTITUTION RULE SYMMETRY Suppose f is continuous on [-a, a] and even even Suppose f is continuous on [-a, a] and odd Odd

  3. Sec 5.5: THE SUBSTITUTION RULE Term-102

  4. Sec 5.5: THE SUBSTITUTION RULE Term-091 (even) + (even) = (even) (odd) +(odd ) = (odd ) (even) X (even) = (even) (odd ) X (odd ) = (even) (even) X (odd ) = (odd )

  5. Sec 5.5: THE SUBSTITUTION RULE Term-103

  6. Sec 5.5: THE SUBSTITUTION RULE SYMMETRY Suppose f is continuous on [-a, a] and even Suppose f is continuous on [-a, a] and odd Example

  7. Sec 5.5: THE SUBSTITUTION RULE T-121

  8. Sec 5.5: THE SUBSTITUTION RULE T-141

  9. Sec 5.5 THE SUBSTITUTION RULE

  10. Sec 5.5: THE SUBSTITUTION RULE Find Table Indefinite Integrals Find Find The Substitution Rule

  11. Sec 5.5: THE SUBSTITUTION RULE Definite Integral evaluate the expression in u Example return to the variable x

  12. Sec 5.5: THE SUBSTITUTION RULE TableIndefiniteIntegrals Find The Substitution Rule

  13. Sec 5.5: THE SUBSTITUTION RULE

  14. Sec 5.5: THE SUBSTITUTION RULE Main Idea In general, this method works whenever we have an integral that written as a product of function and its derivative. & & & & & & & Simple Integral The idea behind the Substitution Rule is to replace a relatively complicated integral by a simpler integral. This is accomplished by changing from the original variable to a new variable that is a function of . Main Challenge The main challenge in using the Substitution Rule is to think of an appropriate substitution. You should try to choose to be some function in the integrand whose differential also occurs (except for a constant factor). Try Another Finding the right substitution is a bit of an art. It’s not unusual to guess wrong; if your first guess doesn’t work, try another substitution.

  15. Sec 5.5: THE SUBSTITUTION RULE Poly. with high power & its derivative u = poly 132

  16. Sec 5.5: THE SUBSTITUTION RULE Poly. In denominator and its derivative in the nominator u = poly 082

  17. Sec 5.5: THE SUBSTITUTION RULE f (sin x) & cos x dx u = sin f (cos x) & sin x dx u = sin 092 141

  18. Sec 5.5: THE SUBSTITUTION RULE ln(x)& 1/x dx u = ln(x) T-102

  19. Sec 5.5: THE SUBSTITUTION RULE tan(x)& sec^2(x) dx u = tan(x) 082

  20. Sec 5.5: THE SUBSTITUTION RULE e^x & e^x dx a^x & a^x dx 121 092

  21. Sec 5.5: THE SUBSTITUTION RULE Sqrt(---)

  22. Sec 5.5: THE SUBSTITUTION RULE & & &

  23. Sec 5.5: THE SUBSTITUTION RULE T-132

  24. Sec 5.5: THE SUBSTITUTION RULE T-132

  25. Sec 5.5: THE SUBSTITUTION RULE T-132

  26. Sec 5.5: THE SUBSTITUTION RULE T-102

  27. Sec 5.5: THE SUBSTITUTION RULE 082

  28. Sec 5.5: THE SUBSTITUTION RULE 082

  29. Sec 5.5: THE SUBSTITUTION RULE 082 T-102

  30. Sec 5.5: THE SUBSTITUTION RULE Find Example Example

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