# How to prove the definition of arctangent by G. H. Hardy through integral?

From introduction to analysis,by Arthur P. Mattuck,problem 20-1. I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks.

Problems 20-1
One way of rigorously defining the trigonometric functions is to start with the definition of the arctangent function. (This is the route used for example in the classic text Pure Mathematics by G. H. Hardy.) So, assume amnesia has wiped out the trigonometric functions (but the rest of your knowledge of analysis is intact). Define $$T(x)=\int_{0}^{x}\frac{dt}{1+t^{2}}$$

(a) Prove T(x) is defined for all x and odd.
(b) Prove T(x) is continuous and differentiable, and find T'(x).
(c) Prove T(x) is strictly increasing for all x; find where it is convex, where concave, and its points of inflection.
(d) Show T(x) is bounded for all x, and |T(x)| < 2.5, using comparison of integrals. Can you get a better bound?

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Don't you want the integrand to be $1\over 1+t^2$ instead of $1\over 1+x^t$? And perhaps you want to find $T'(x)$ instead of $T(x)$ in (b)? – Per Manne Oct 15 '12 at 16:41
Consider the integral $\int_1^x dt/t^2$. – GEdgar Oct 15 '12 at 16:42
The magic number $2.5$ is quite a bit too large. One can do better, and it is not clear to me how to obtain such a poor bound in a natural way. – André Nicolas Oct 15 '12 at 17:05
It is diffcult to get the loose upper bound 2.5,even through $$\int_{1}^{x}\frac{dt}{t^{2}}$$ .Any more idea? – inix Oct 16 '12 at 14:51
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