Consider the integral $\displaystyle I$ defined by

$\displaystyle \displaystyle I = \int^1_0 e^{-xt} t \cos t dt$.

Use Laplace's method to obtain the first term of the asymptotic expansion of the integral as $\displaystyle x \to \infty$.

How can I start? Can someone point me to somewhere with a similar example?

Thank you.