Prove that limx →∞ ∫1^x (1)/(1+t^2) d t=∫0^1 (1)/(1+t^2) d t=(π)/(4) that is, limx →∞ arctanx=ar

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Prove that limx →∞ ∫1^x (1)/(1+t^2) d t=∫0^1 (1)/(1+t^2) d...

Prove that $$ \lim _{x \rightarrow \infty} \int_{1}^{x} \frac{1}{1+t^{2}} d t=\int_{0}^{1} \frac{1}{1+t^{2}} d t=\frac{\pi}{4} $$ that is, $\lim _{x \rightarrow \infty} \arctan x=\arctan 1=\pi / 4$. Deduce that $2.88<\pi<3.39$. (Hint: Substitute $t=1 / s$ and use Proposition 6.20. Divide $[0,1]$ into subintervals of length $\frac{1}{4}$.)

Prove that
$$
\lim _{x \rightarrow \infty} \int_{1}^{x} \frac{1}{1+t^{2}} d t=\int_{0}^{1} \frac{1}{1+t^{2}} d t=\frac{\pi}{4}
$$
that is, $\lim _{x \rightarrow \infty} \arctan x=\arctan 1=\pi / 4$. Deduce that $2.88<\pi<3.39$.
(Hint: Substitute $t=1 / s$ and use Proposition 6.20. Divide $[0,1]$ into subintervals of length $\frac{1}{4}$.)

Key Concepts

Limits and Behavior of Inverse Trigonometric Functions

Understanding the limits of functions, especially inverse trigonometric ones like arctan, is critical in evaluating improper integrals. Here, recognizing that the limit of arctan x as x approaches infinity equals ?/2 is pivotal for comparing certain integral expressions and deducing bounds for ?.

Fundamental Theorem of Calculus

This theorem bridges the concepts of differentiation and integration by stating that if a function is continuous over an interval, then its definite integral over that interval can be computed using an antiderivative. In the given problem, it is used to represent the integral of 1/(1+t^2) as the arctan function, establishing the equality between the integral and arctan x evaluated at the endpoints.

Substitution in Integration

Substitution is an essential technique in integration whereby a change of variable simplifies the integrand or the limits of integration. In this problem, the substitution t = 1/s is suggested as a means to transform the integral into a form that can be more easily analyzed, particularly to explore the symmetry or behavior of the integrand over a new interval.

Numerical Integration and Estimation

Dividing an interval into subintervals and approximating the area under a curve is a common method for estimating the value of a definite integral. In the context of this problem, partitioning the interval [0,1] assists in providing numerical bounds for the integral, which in turn leads to deducing the inequalities for the value of ?.

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