The value of the integral $\underset{-\infty }{\overset{\infty }{\int }}\frac{dx}{1+x^2}$ is

Integral Evaluation Explained

The question asks for the value of the definite integral of the function $\frac{1}{1+x^2}$ over the entire real line, from negative infinity to positive infinity. This is a common integral in calculus.

The integral we need to evaluate is given by: $$\underset{-\infty }{\overset{\infty }{\int }}\frac{dx}{1+x^2}$$

Antiderivative of the Function

To solve this definite integral, we first need to find the antiderivative of the integrand $\frac{1}{1+x^2}$.

- We know that the derivative of the inverse tangent function, $\arctan (x)$ (also written as ${\tan }^{-1}(x)$), is $\frac{1}{1+x^2}$.
- Therefore, the antiderivative of $\frac{1}{1+x^2}$ with respect to $x$ is $\arctan (x)$.

Evaluating the Definite Integral at Infinity

Now, we evaluate the definite integral using the fundamental theorem of calculus for improper integrals. This involves taking limits as $x$ approaches infinity and negative infinity.

The evaluation proceeds as follows: $$\underset{-\infty }{\overset{\infty }{\int }}\frac{dx}{1+x^2}={\left[\arctan (x)\right]}_{-\infty }^{\infty }$$ This can be written as:
$$\underset{b\to \infty }{\lim }\arctan (b)-\underset{a\to -\infty }{\lim }\arctan (a)$$

- **Limit as $x$ approaches positive infinity:** As $x$ approaches positive infinity, the value of $\arctan (x)$ approaches $\frac{\pi }{2}$. $$\underset{x\to \infty }{\lim }\arctan (x)=\frac{\pi }{2}$$
- **Limit as $x$ approaches negative infinity:** As $x$ approaches negative infinity, the value of $\arctan (x)$ approaches $-\frac{\pi }{2}$. $$\underset{x\to -\infty }{\lim }\arctan (x)=-\frac{\pi }{2}$$

Calculating the Final Value of the Integral

Substitute these limit values back into the expression for the definite integral:
$$\frac{\pi }{2}-\left(-\frac{\pi }{2}\right)$$

$$=\frac{\pi }{2}+\frac{\pi }{2}$$

$$=\pi$$

Thus, the value of the integral $\underset{-\infty }{\overset{\infty }{\int }}\frac{dx}{1+x^2}$ is $\pi$. This integral is a classic result in calculus, often associated with the calculation of areas related to the Cauchy distribution or probability.