Integration and Carbon-14 help

Questions:

Integrate e^(x)sec(e^(x))dx

Carbon 14 question: How old is a painting when it has 99.5% of it's original carbon 14? (assuming 5700 is the half-life of carbon 14)

I think you use u-substitution for the 1st one, but I'm not so sure what to do with that, so walk me through it. Also the answer for that is ln lsec(e^(x)+tan(e^(x))l+c (read as logarithm of the absolute value...etc)

For the other one, am I supposed to memorize the constant for carbon-14? Thank you