766
Appendix A
2 −1 4 cos
−1 u − u 1− u 2
63. ∫ u cos −1 udu = 2 u
C
4 +
2 +1
64. ∫ u tan −1 udu = u
−1 u − u
C
2 tan
2 +
⎡ ⎣ ⎢ u n +1 sin −1 u − ∫ u
⎤ ⎦ ⎥ , n ≠−1
n +1 du 1− u 2
65. ∫ u n sin −1 udu = 1 n +1
⎡ ⎣ ⎢ u n +1 cos −1 u + ∫ u
⎤ ⎦ ⎥ , n ≠−1
n +1 du 1− u 2
66. ∫ u n cos −1 udu = 1 n +1
67. ∫ u n tan −1 udu = 1 n +1 ⎡ ⎤ ⎦ , n ≠−1 Integrals Involving a 2 + u 2 , a > 0 68. ∫ a 2 + u 2 du = u 2 a 2 + u 2 + a 2 2 ln ⎛ ⎝ u + a 2 + u 2 ⎞ ⎠ + C 69. ∫ u 2 a 2 + u 2 du = u 8 ⎛ ⎝ a 2 +2 u 2 ⎞ ⎠ a 2 + u 2 − a 4 8 ln ⎛ ⎝ u + a 2 + u 2 ⎞ ⎠ + C 70. ∫ a 2 + u 2 u du = a 2 + u 2 − a ln | a + a 2 + u 2 u | + C 71. ∫ a 2 + u 2 u 2 du = − a 2 + u 2 u +ln ⎛ ⎝ u + a 2 + u 2 ⎞ ⎠ + C 72. ∫ du a 2 + u 2 = ln ⎛ ⎝ u + a 2 + u 2 ⎞ ⎠ + C 73. ∫ u 2 du a 2 + u 2 = u 2 ⎛ ⎝ a 2 + u 2 ⎞ ⎠ − a 2 2 ln ⎛ ⎝ u + a 2 + u 2 ⎞ ⎠ + C 74. ∫ du u a 2 + u 2 = − 1 a ln | a 2 + u 2 + a u | + C 75. ∫ du u 2 a 2 + u 2 = − a 2 + u 2 a 2 u + C 76. ∫ du ⎛ ⎝ a 2 + u 2 ⎣ u n +1 tan −1 u − ∫ u n +1 du 1+ u 2 ⎞ ⎠ 3/2 = u a 2 a 2 + u 2 + C Integrals Involving u 2 − a 2 , a > 0 77. ∫ u 2 − a 2 du = u 2 u 2 − a 2 − a 2 2 ln | u + u 2 − a 2 | + C 78. ∫ u 2 u 2 − a 2 du = u 8 ⎛ ⎝ 2 u 2 − a 2 ⎞ ⎠ u 2 − a 2 − a 4 8 ln | u + u 2 − a 2 | + C
2 − a 2 u du = u
79. ∫ u
2 − a 2 − a cos −1 a
C
| u | +
+ln | u + u
2 − a 2 | + C
2 − a 2 u 2
2 − a 2 u
80. ∫ u
du = − u
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