Question

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

Answer 1

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

Use integration by parts 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.