I am reading something on the Laplace method of integrals and I don't understand part of it's argument. It gives the integral
∫4−3e−λx2log(1+x2)dx
and finding the leading term of asymptotics for λ→∞. It first argues
"Since only small |x|, such that |x|∼1√λ≪1 are important,
log(1+x2)∼x2
I(\lambda)\sim \int_{-3}^4 e^{-\lambda x^2}x^2 \, dx \sim \int_{-\infty}^{\infty} e^{-\lambda x^2}x^2 \, dx \sim 2\int_{0}^{\infty} e^{-\lambda x^2}x^2 \, dx."
I'm really not following that argument at all. Any help would be appreciated.
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