Let f be continuous on [0,1] and J1=∫0^1 / √(π) (n f(x))/(1+n^2 x^2) d x J2=∫1 / √(n)

step 1 Question we have to prove integration of function of cos x d x the limit is 0 to pi by 2 is equa

Let f be continuous on [0,1] and J1=∫0^1 / √(π) (n...

Let $f$ be continuous on $[0,1]$ and $$ \begin{aligned} &J_{1}=\int_{0}^{1 / \sqrt{\pi}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \\ &J_{2}=\int_{1 / \sqrt{n}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \end{aligned} $$ Use 7.1.12 to prove that $$ \begin{aligned} &J_{1}=f\left(c_{n}\right) \tan ^{-1} \sqrt{n} \\ &J_{2}=f\left(d_{n}\right)\left(\tan ^{-1} n-\tan ^{-1} \sqrt{n}\right) \end{aligned} $$ where $0<c_{n}<1 / \sqrt{n}<d_{n}<1$. Deduce that $$ \int_{0}^{1} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \rightarrow \frac{1}{2} \pi f(0) \quad \text { as } n \rightarrow x $$

Let $f$ be continuous on $[0,1]$ and
$$
\begin{aligned}
&J_{1}=\int_{0}^{1 / \sqrt{\pi}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \\
&J_{2}=\int_{1 / \sqrt{n}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x
\end{aligned}
$$
Use 7.1.12 to prove that
$$
\begin{aligned}
&J_{1}=f\left(c_{n}\right) \tan ^{-1} \sqrt{n} \\
&J_{2}=f\left(d_{n}\right)\left(\tan ^{-1} n-\tan ^{-1} \sqrt{n}\right)
\end{aligned}
$$
where $0<c_{n}<1 / \sqrt{n}<d_{n}<1$. Deduce that
$$
\int_{0}^{1} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \rightarrow \frac{1}{2} \pi f(0) \quad \text { as } n \rightarrow x
$$

Key Concepts

Mean Value Theorem for Integrals

This theorem states that if a function is continuous on a closed interval, then there exists at least one point in the interval such that the integral of the function over the interval is equal to the function’s value at that point multiplied by the length (or measure) of the interval. It provides a way to replace an integral with an expression involving the function’s value at a specific point, which is crucial when estimating or evaluating parameter?dependent integrals.

Continuity

Continuity of a function on a closed interval ensures that the function does not have any abrupt changes and that it attains every value between its minimum and maximum. This property is essential for applying the Mean Value Theorem for Integrals and for ensuring that limits involving the function's values are well-behaved, especially when the integral is influenced by a parameter that causes concentration of the integrand.

Asymptotic Analysis in Parameter-Dependent Integrals

Asymptotic analysis involves studying the behavior of integrals and other expressions as a parameter becomes large or small. In integrals that depend on a parameter, such analysis helps in understanding how the weighting of the integrand changes, and in determining the limiting behavior of the integral. This is particularly important when the integrand becomes highly concentrated in a small region as the parameter increases.

Approximate Identities in Integration

An approximate identity is a family of functions that, as a parameter tends to a limit, behaves like a Dirac delta function, concentrating the mass of the function at a specific point. In the context of integration, such families act to 'select' the function’s value near that point, which is a key technique in analyzing limits of integrals where the weight function becomes more concentrated around a point as the parameter varies.

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