Integrate

Question

cosh^2x?

 

I get up to the step

1/4 sinh(2x) + x/2

but then I don't understand how it turns into

x/2 + 1/2 sinh(x)cosh(x)  

 

Answer 1

cosh²x = (1+cosh2x)/2.

∫cosh²x = ∫(1+cosh2x)/2.

              =(1/2)∫(1+cosh2x)

               =(1/2)(x+(1/2)sinh2x)

                =(x/2)+(1/4)(sinh2x)

Here sinh2x=2sinhxcoshx

  =(x/2)+(1/4)(2sinhxcoshx)

 =(x/2)+(1/2)(sinhxcoshx)