∫
csch−^1 xadx=xcsch−^1 xa±asinh−^1 xa, +forx > 0 and−forx < 0∫
xsinh−^1 xadx=(
x^2
2 +a^2
4)
sinh−^1 xa−^14 xx√
x^2 +a^2 +C∫
xcosh−^1 xadx=
1
4 (2x(^2) −a (^2) ) cosh− 1 x
a−
1
4 x
√
x^2 −a^2 +C, cosh−^1 (x/a)> 0
1
4 (2x
(^2) −a (^2) ) cosh− 1 x
a+
1
4 x
√
x^2 −a^2 +C, cosh−^1 (x/a)< 0
667.
∫
xtanh−^1 xadx=ax 2 +^12 (x^2 −a^2 ) tanh−^1 xa+C
668.
∫
xcoth−^1 xadx=ax 2 +^12 (x^2 −a^2 ) coth−^1 xa+C
669.
∫
xsech−^1 xadx=
1
2 x
(^2) sech− 1 x
a−
1
2 a
√
a^2 −x^2 , sech−^1 (x/a)> 0
1
2 xsech
− 1 x
a+
1
2 a
√
a^2 −x^2 +C, sech−^1 (x/a)< 0
670.
∫
xcsch−^1 xadx=^12 x^2 csch−^1 xa±a 2
√
x^2 +a^2 +C, +forx > 0 and−forx < 0
671.
∫
x^2 sinh−^1 xadx=^13 x^3 sinh−^1 xa+^19 (2a^2 −x^2 )
√
x^2 +a^2 +C
672.
∫
x^2 cosh−^1 xadx=
1
3 x
(^3) cosh−^1 x
a−
1
9 (x
(^2) + 2a (^2) )√x (^2) −a (^2) +C, cosh−^1 (x/a)> 0
1
3 x
(^3) cosh−^1 x
a+
1
9 (x
(^2) + 2a (^2) )√x (^2) −a (^2) +C, cosh−^1 (x/a)< 0
673.
∫
x^2 tanh−^1 xadx=a 6 x^2 +^13 x^3 tanh−^1 xa+^16 a^3 ln|a^2 −x^2 |+C
674.
∫
x^2 coth−^1 xadx=a 6 x^2 +^13 x^3 coth−^1 xa+^16 a^3 ln|x^2 −a^2 |+C
675.
∫
x^2 sech−^1 xadx=^13 x^3 sech−^1 xa−^13
∫ x (^3) dx
√
x^2 +a^2
676.
∫
x^2 csch−^1 xadx=^13 x^3 csch−^1 xa±a 3
∫ x (^2) dx
√
x^2 +a^2
677.
∫
xnsinh−^1 xadx=n+ 1^1 xn+1sinh−^1 xa−n+ 1^1
∫ xn+1dx
√
x^2 −a^2
678.
∫
xncosh−^1 xadx=
1
n+ 1x
n+1cosh−^1 x
a−
1
n+ 1
∫ xn+1
√
x^2 −a^2
, cosh−^1 (x/a)> 0
1
n+ 1x
n+1cosh− 1 x
a+
1
n+ 1
∫ xn+1dx
√
x^2 −a^2
, cosh−^1 (x/a)< 0
679.
∫
xntanh−^1 xadx=n+ 1^1 xn+1tanh−^1 xa−n+ 1a
∫ xn+1dx
a^2 −x^2
Appendix C