`int cs ch(1/x)coth(1/x)/x^2 dx` Find the indefinite integral

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Those  `1/x`  under the hyperbolic cosecant and cotangent are irritating, let's change them to more appropriate `y.` Make the substitution `1/x = y,` then `x = 1/y` and `dx = -1/y^2 dy.` The integral becomes

`-int (cs ch(y) coth(y))/(1/y^2) (dy)/y^2 = -int cs ch(y) coth(y) dy =`

|recall that  `cs ch(y) = 1/sinh(y)`  and  `coth(y) = cosh(y)/sinh(y)` |

`= -int cosh(y)/(sinh^2(y)) dy.`

The next substitution is `u = sinh(y),` then `du = cosh(y) dy,` and the integral becomes

`- int (du)/u^2 = 1/u + C = 1/sinh(y) + C = 1/sinh(1/x) + C = cs ch(1/x) + C,`

where `C` is an arbitrary constant.

Approved by eNotes Editorial Team