cos t

1

[0, 3 π]

at the stationary endpoint

t = 0

and at the interior point

t = 2 π

. At either point,

cos (t_{0} + u) = 1 - \frac{u ^{2}}{2} + O (u^{4}) .

e^{x} \frac{π}{2 x} .

By Laplace's method, the interior maximum contributes

e^{x + 2 π} \frac{2 π}{x} .

The remaining parts of the interval are exponentially smaller. Summing the two contributions according to Laplace's method with multiple global maxima gives

I (x) \sim e^{x} \frac{π}{2 x} (1 + 2 e^{2 π}) .

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