Paper 3, Section II, D

Asymptotic Methods | Part II, 2020

(a) Find the leading order term of the asymptotic expansion, as x→∞x \rightarrow \infty, of the integral

I(x)=∫03πe(t+xcos⁡t)dtI(x)=\int_{0}^{3 \pi} e^{(t+x \cos t)} d t

(b) Find the first two leading nonzero terms of the asymptotic expansion, as x→∞x \rightarrow \infty, of the integral

J(x)=∫0π(1−cos⁡t)e−xln⁡(1+t)dtJ(x)=\int_{0}^{\pi}(1-\cos t) e^{-x \ln (1+t)} d t

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