Bourbaki

Exercise 33, § 1

Functions of a Real Variable, III, 123

a) Calculate the integrals

∫01(1−x2)ndx,∫0+∞dx(1+x2)n \int_0^1 (1 - x^2)^n dx, \qquad \int_0^{+\infty} \frac{dx}{\left(1 + x^2\right)^n}
for nn an integer >0> 0, with the help of exerc. 31. b) Show that one has
1−x2≤e−x2for 0≤x≤1e−x2≤11+x2for x≥0. 1 - x^2 \leq e^{-x^2} \qquad \text{for } 0 \leq x \leq 1 \\ e^{-x^2} \leq \frac{1}{1 + x^2} \qquad \text{for } x \geq 0.
c) Deduce from a) and b) and from Wallis' formula (exerc. 32) that
∫0+∞e−x2dx=π2. \int_0^{+\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}.

Solution

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Transcribed from the printed volume. Not translated.