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

1 Answer
Jun 7, 2015

\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.