How do you find the integral of ##(cosx)(coshx) dx##?

##\int cos(x)cosh(x)dx=[cos(x)sinh(x)+cosh(x)sin(x)]/2+C##

Use twice

Using integration by parts once we get,

##\int cos(x)cosh(x)dx=cos(x)sinh(x)+\intsinh(x)sin(x)dx##

Now use integration by parts on the right hand integral

##int sinh(x)sin(x)dx=cosh(x)sin(x)-intcos(x)cosh(x)dx##

Substitute this into the top equation and rearrange to get

##2\int cos(x)cosh(x)dx=cos(x)sinh(x)+cosh(x)sin(x)##

##\int cos(x)cosh(x)dx=[cos(x)sinh(x)+cosh(x)sin(x)]/2+C##

Where ##C## is the constant of integration. When applying integration by parts to an indefinite integral, we pick up a constant of integration. I just neglected to write it until the end.