Question

prove that integral of sin^2wdw/w^2 having limits 0 to infinite =pi/2 ??

Answer 1

Conditions

prove that integral of $\sin^2 w \mathrm{d}w / w^2$ having limits 0 to infinite =pi/2 ??

Solution

Consider the integral:

$$\int_{0}^{\infty} \frac{\sin^2 w \, dw}{w^2}$$

As we know, this integral cannot be represented by using elementary functions, that's why in math appeared the definition of Sine integral:

$$\operatorname{Si}(x) = \int_{0}^{x} \frac{\sin t}{t} \, dt$$

One of the properties of this integral is:

$$\lim_{x \to +\infty} \operatorname{Si} x = \frac{\pi}{2},$$

It's known, that

$$\int \frac{\sin^2 w \, dw}{w^2} = \operatorname{Si}(2w) - \frac{\sin^2 w}{w} + c$$ $$\int_{0}^{\infty} \frac{\sin^2 w \, dw}{w^2} = \lim_{x \to \infty} \int_{0}^{x} \frac{\sin^2 w \, dw}{w^2} = \lim_{x \to \infty} \operatorname{Si}(2x) - 0 - 0 + 0 = \frac{\pi}{2}$$