Travel expenses through calculation & invoice

I had to visit sites for a customer and incurred expenses such as flights and hotels. These expenses are reimbursed by the customer.
I suspect that it is best not to include them in my invoice to the customer, as they would then be considered income? And also taxable… Or should I put them in a separate invoice and also list them as my own expenses? I suspect that it is best not to include them in the invoice and just have the customer pay the expenses?