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Integrals With Only Even Powers cos 2 xdx Integrals With Only Even Powers cos 2 xdx Here we need to use the trig identity: Integrals With Only Even Powers cos 2 xdx Here we need to use the trig identity: 1 + cos cos 2

  2. Integrals With Only Even Powers � cos 2 xdx

  3. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity:

  4. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2

  5. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that:

  6. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2

  7. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2 And if we square both sides:

  8. Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2 And if we square both sides: cos 2 x = 1 + cos 2 x 2

  9. Integrals With Only Even Powers So, we replace in our integral:

  10. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2

  11. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � cos 2 xdx =

  12. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � � cos 2 xdx =

  13. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � cos 2 xdx = 2

  14. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � cos 2 xdx = dx 2

  15. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � cos 2 xdx = dx = 2

  16. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2

  17. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � =

  18. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 = 2

  19. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 = 2 dx +

  20. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � = 2 dx +

  21. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x = 2 dx + 2

  22. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x = 2 dx + dx 2

  23. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x dx = 1 = 2 dx + 2 2

  24. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 = 2 dx + 2 2

  25. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 = 2 dx + dx + 2 2

  26. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 dx + 1 = 2 dx + 2 2 2

  27. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + 2 2 2

  28. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + cos 2 x 2 2 2

  29. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + cos 2 xdx 2 2 2

  30. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x

  31. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x = x 2+

  32. Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x � = x 2 + 1 cos 2 xdx 2

  33. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2

  34. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution:

  35. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u =

  36. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u = 2 x

  37. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u = 2 x ,

  38. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx =

  39. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2

  40. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 x 2+

  41. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 x 2 + 1 2

  42. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � x 2 + 1 2

  43. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � x 2 + 1 cos u 2

  44. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du cos u . 2 2 dx

  45. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du cos u . dx 2 2 dx

  46. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚

  47. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ = x 2+

  48. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ 2 + 1 = x 4

  49. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 4

  50. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � 2 + 1 x du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos u 4

  51. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � 2 + 1 x du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu 4

  52. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu 4 � �� � sin u

  53. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu = x 4 2

  54. Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu = x 4 2

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