example 1

Example 1 ln x x dx Example 1 ln x x dx We make the - PowerPoint PPT Presentation

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Example 1 ln x x dx Example 1 ln x x dx We make the substitution: Example 1 ln x x dx We make the substitution: u = ln x Example 1 ln x x dx We make the substitution: du dx = 1 u = ln x , x Example 1 ln x x dx We

  2. Example 1 � ln x x dx

  3. Example 1 � ln x x dx We make the substitution:

  4. Example 1 � ln x x dx We make the substitution: u = ln x

  5. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x

  6. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x x dx =

  7. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x � x dx = u

  8. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x u . du � x dx = dx

  9. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x u . du � x dx = dx dx

  10. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x u . du � ✚ x dx = dx ✚ dx ✚ ✚

  11. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x u . du � � ✚ x dx = dx ✚ dx = udu ✚ ✚

  12. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x udu = u 2 � u . du � ✚ dx ✚ x dx = dx = 2 + C ✚ ✚

  13. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x udu = u 2 � u . du � ✚ dx ✚ x dx = dx = 2 + C ✚ ✚ Now we substitute back:

  14. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x udu = u 2 � u . du � ✚ dx ✚ x dx = dx = 2 + C ✚ ✚ Now we substitute back: � ln x x dx =

  15. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x udu = u 2 � u . du � ✚ dx ✚ x dx = dx = 2 + C ✚ ✚ Now we substitute back: � ln x x dx = (ln x ) 2 + C 2

  16. Example 1 � ln x x dx We make the substitution: du dx = 1 u = ln x , x � ln x udu = u 2 � u . du � ✚ dx ✚ x dx = dx = 2 + C ✚ ✚ Now we substitute back: � ln x x dx = (ln x ) 2 + C 2

  17. Example 2 e x � 1 + e 2 x dx

  18. Example 2 e x � 1 + e 2 x dx We can write this integral as:

  19. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x 1 + e 2 x dx =

  20. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2

  21. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution:

  22. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: u = e x ,

  23. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x

  24. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x � 1 + ( e x ) 2 =

  25. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 � � 1 + ( e x ) 2 = 1 + u 2 .

  26. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 + u 2 . du 1 � � 1 + ( e x ) 2 = dx

  27. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 + u 2 . du 1 � � 1 + ( e x ) 2 = dx dx

  28. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 . du � � 1 + ( e x ) 2 = dx dx e 2 x � ✒ 1 + � u 2

  29. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x ✁ ✕ � e x � 1 du ✁ 1 + ( e x ) 2 = dx dx 1 + u 2 . ✁

  30. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 + u 2 . du 1 � � 1 + ( e x ) 2 = dx dx

  31. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 + u 2 . du 1 � � ✚ 1 + ( e x ) 2 = dx ✚ dx ✚ ✚

  32. Example 2 e x � 1 + e 2 x dx We can write this integral as: � e x � e x 1 + e 2 x dx = 1 + ( e x ) 2 So, we make the substitution: du u = e x , dx = e x e x 1 + u 2 . du 1 du � � � ✚ 1 + ( e x ) 2 = dx ✚ dx = ✚ ✚ 1 + u 2

  33. Example 2

  34. Example 2 du � 1 + u 2 =

  35. Example 2 du � 1 + u 2 = arctan u + C

  36. Example 2 du � 1 + u 2 = arctan u + C Substituting back:

  37. Example 2 du � 1 + u 2 = arctan u + C Substituting back: u = e x

  38. Example 2 du � 1 + u 2 = arctan u + C Substituting back: u = e x � e x 1 + e 2 x dx =

  39. Example 2 du � 1 + u 2 = arctan u + C Substituting back: u = e x � e x 1 + e 2 x dx = arctan e x + C

  40. Example 2 du � 1 + u 2 = arctan u + C Substituting back: u = e x � e x 1 + e 2 x dx = arctan e x + C

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