The integral has an integrand that grows without bound at

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Computational Physics, Ch Differentiating and integrating numbers

Problem

The integral

I=\int_0^1 x^{-1/2}e^{-x}\,\mathrm{d}x

has an integrand that grows without bound at

x=0

, yet the area under it is finite. Its exact value is

I=\sqrt{\pi}\,\mathrm{erf}(1)=1.4936483

. ((a)) Say why the trapezium rule and Simpson's rule cannot be run on

I

as it stands, and why the composite midpoint rule can be. State the order a halving table for the composite midpoint rule would show on this integrand, and why it is not

2

. ((b)) Find the substitution that removes the singularity, and show that it turns

I

into

2\int_0^1 e^{-u^2}\,\mathrm{d}u

, whose integrand is bounded and smooth on the whole interval. ((c)) Apply Simpson's rule to the transformed integral at

N=4

8

and

16

, quote each estimate to seven figures, and form the ratios of consecutive errors against

I

. State whether the order is the one Simpson's derivation promises.

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