Math documents

Integral of sech x


Unlike the trigonometric functions  sec x and csc x  and the hyperbolic function  csch x, the function  sech x  can be integrated in more than one way, with each method producing a different result. These results are useful because they "connect" the hyperbolic functions to the trigonometric functions without the use of complex math - a discovery made by Christoph Gudermann.

 

Method 1

sech x dx = 
1
cosh x
 dx =   
cosh x
cosh2x
 dx = 
cosh x
1+sinh2x
 dx

Let U = sinh x. Then dU = cosh x dx, and the integral becomes

sech x dx = 
dU
1 + U2
 = tan-1 U + C

sech x dx = tan-1(sinh x) + C

Using the identity  cot-1 x = tan-1 (1/x), the integral is also

sech x dx = cot-1(csch x) + C

Method 2

sech x dx =   
sech2x
sech x
 dx = 
sech2x
 
1-tanh2x
 dx

Let U = tanh x. Then dU = sech2 x dx, and the integral becomes

sech x dx = 
dU
 
1 - U2
 = sin-1 U + C

sech x dx = sin-1(tanh x) + C

Using the identity  csc-1 x = sin-1 (1/x), the integral is also

sech x dx = csc-1(coth x) + C


Method 3

sech x dx =   
sech x tanh x
tanh x
 dx =   
sech x tanh x
 
1 - sech2 x
 dx

Let U = sech x. Then dU = -sech x tanh x dx, and the integral becomes

sech x dx = 
-dU
 
1 - U2
 = cos-1 U + C

sech x dx = cos-1(sech x) + C

Using the identity  sec-1 x = cos-1 (1/x), the integral is also

sech x dx = sec-1(cosh x) + C


Method 4

sech x dx = 
1
cosh x
 dx

Apply the double-"angle" formula  cosh 2U = cosh2 U + sinh2 U  to cosh x:

sech x dx 
1
cosh2 (
x
2
)  + sinh2 (
x
2
)
 dx
 
 
1
cosh2 (
x
2
)
( 1 + tanh2 (
x
2
) )
 dx
 
 
sech2 (
x
2
)
1 + tanh2 (
x
2
)
 dx

Let U = tanh(x/2). Then dU = ½ sech2(x/2) dx, and the integral becomes

sech x dx = 2
dU
1 + U2
 = 2 tan-1 U + C

sech x dx = 2 tan-1
( tanh (
x
2
) )
 + C

Method using the sech x definition

sech x dx = 
2
ex + e-x
 dx = 2
ex
e2x + 1
 dx

Let U = ex. Then dU = ex dx, and the integral becomes

sech x dx = 2
dU
U2 + 1
 = 2 tan-1 U + C = 2 tan-1
( ex )
 + C


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