∫ dx
α+βsinhax=
√^1
α^2 +β^2ln∣∣
∣∣
∣βeax+α−√
α^2 +β^2
βeax+α+√
α^2 +β^2∣∣
∣∣
∣+C∫ dx
(α+βsinhax)^2 =−β
a(α^2 +β^2 )coshax
α+βsinhax+α
α^2 +β^2∫ dx
α+βsinhax∫ dx
α^2 +β^2 sinh^2 ax=
1
aα√
β^2 −α^2tan−^1(√
β^2 −α^2 tanhax
α)
+C, β^2 > α^21
2 aα√
α^2 −β^2ln∣∣
∣∣
∣α+√
α^2 −β^2 tanhax
α−√
α^2 −β^2 tanhax∣∣
∣∣
∣+C, β(^2) < α 2
604.
∫ dx
α^2 −β^2 sinh^2 ax
=^1
2 aα
√
α^2 +β^2
ln
∣∣
∣∣
∣
α+
√
α^2 +β^2 tanhax
α−
√
α^2 +β^2 tanhax
∣∣
∣∣
∣+C
Integrals containing the hyperbolic function coshax
605.
∫
coshax dx=^1 asinhax+C
606.
∫
xcoshax dx=^1 axsinhax−a^12 coshax+C
607.
∫
x^2 coshax dx=−a^22 xcoshax+
(x 2
a +
2
a^3
)
sinhax+C
608.
∫
xncoshax dx=a^1 xnsinhax−na
∫
xn−^1 sinhax dx
609.
∫ 1
xcoshax dx= ln|x|+
(ax)^2
2 ·2!+
(ax)^4
4 ·4!+
(ax)^6
6 ·6!+···+C
610.
∫ 1
x^2 coshax dx=−
1
xcoshax+a
∫ 1
xsinhax dx
611.
∫ 1
xncoshax dx=−
1
n− 1
coshax
xn−^1 +
a
n− 1
∫ sinhax
xn−^1 dx, n >^1
612.
∫ dx
coshax=
2
atan
− (^1) eax+C
613.
∫ x dx
coshax=
1
a^2
[a (^2) x 2
2 −
a^4 x^4
8 +
5 a^6 x^6
144 +···+ (−1)
nEna^2 n+2x^2 n+2
(2n+ 2)·(2n)!+···
]
+C
614.
∫
cosh^2 ax dx=^12 x+^12 sinhaxcoshax+C
615.
∫
coshnax dx=na^1 coshn−^1 axsinhax+n−n^1
∫
coshn−^2 ax dx
616.
∫
xcosh^2 ax dx=^14 x^2 + 41 axsinh2ax− 8 a^12 cosh 2ax+C
Appendix C