∫
xtan−^1 xadx=^12 (x^2 +a^2 ) tan−^1 xa−a 2 ln|x^2 +a^2 |+C∫
xcot−^1 xadx=^12 (x^2 +a^2 ) cot−^1 xa+a 2 x+C∫
xsec−^1 xadx=
1
2 x(^2) sec− 1 x
a−
a
2
√
x^2 −a^2 +C, 0 <sec−^1 xa< π/ 2
1
2 x
(^2) sec− 1 x
a+
a
2
√
x^2 −a^2 +C, π/ 2 <sec−^1 xa< π
526.
∫
xcsc−^1 xadx=
1
2 x
(^2) csc− 1 x
a+
a
2
√
x^2 −a^2 +C, 0 <csc−^1 xa< π/ 2
1
2 x
(^2) csc− 1 x
a−
a
2
√
x^2 −a^2 +C, −π/ 2 <csc−^1 xa< 0
527.
∫
x^2 sin−^1 xadx=^13 x^3 sin−^1 xa+^19 (x^2 + 2a^2 )
√
a^2 −x^2 +C
528.
∫
x^2 cos−^1 xadx=^13 x^3 cos−^1 xa−^19 (x^2 + 2a^2 )
√
a^2 −x^2 +C
529.
∫
x^2 tan−^1 xadx=^13 tan−^1 xa−a 6 x^2 +a
3
6 ln|x
(^2) +a (^2) |+C
530.
∫
x^2 cot−^1 xadx=^13 cot−^1 xa+a 6 x^2 −a
3
6 ln|a
(^2) +x (^2) |+C
531.
∫
x^2 sec−^1 xadx=
1
3 x
(^3) sec− 1 x
a−
a
6 x
√
x^2 −a^2 −a
3
6 ln|x+
√
x^2 −a^2 |+C, 0 <sec−^1 xa< π/ 2
1
3 x
(^3) sec− 1 x
a+
a
6 x
√
x^2 −a^2 +a
3
6 ln|x+
√
x^2 −a^2 |+c, π/ 2 <sec−^1 xa< π
532.
∫
x^2 csc−^1 xadx=
1
3 x
(^3) csc− 1 x
a+
a
6 x
√
x^2 −a^2 +a
3
6 ln|x+
√
x^2 −a^2 |+C, 0 <csc−^1 xa< π/ 2
1
3 x
(^3) csc− 1 x
a−
a
6 x
√
x^2 −a^2 −a
3
6 ln|x+
√
x^2 −a^2 |+C, −π/ 2 <csc−^1 xa< 0
533.
∫ 1
xsin
− 1 x
adx=
x
a+
1
2 · 3 · 3
(x
a
) 3
- 2 ·^14 ··^35 · 5
(x
a
) 5 - 2 ·^14 ··^36 ··^57 · 7 +···+C
- ∫ 1
xcos
− 1 x
adx=
π
2 ln|x|+−
∫ 1
xsin
− 1 x
adx
∫ 1
xtan
− 1 x
adx=
x
a−
1
32
(x
a
) 3
- 512
(x
a
) 5
− 712
(x
a
) 7
+···+C
- ∫ 1
xcot
− 1 x
adx=
π
2 ln|x|−
∫ 1
xtan
− 1 x
adx
∫ 1
xsec
− 1 x
adx=
π
2 ln|x|+
a
x+
1
2 · 3 · 3
(x
a
) 3
- 2 ·^14 ··^35 · 5
(x
a
) 5 - 2 ·^14 ·· 63 ··^57 · 7
(x
a
) 7
+···+C
Appendix C