# Evaluating the integral $\int_0^{\infty} \cos(x^2)\, \mathrm{d} x$?

Is it necessary to make use of the Gaussian integral and the complex exponential form of the cosine in evaluating the following integral?

$$\int_0^{\infty} \cos(x^2)\, \mathrm{d} x$$

Just curious - I can prove its convergence, but not evaluate it at the moment.

## 1 Answer

Is it necessary to use the Gamma function in integral form and then make the appropriate changes.... probably not, but is one of the shortest methods to choose.

Consider the integral \begin{align} I_{n} = \int_{0}^{\infty} \cos(x^{2n}) \, dx \end{align} and make the change of variables $t = x^{2n}$ which leads to \begin{align} I_{n} = \frac{1}{2n} \, \int_{0}^{\infty} \cos(t) \, t^{\frac{1}{2n} - 1} \, dt. \end{align} Now using the integral \begin{align} \int_{0}^{\infty} \cos(at) \, t^{p-1} \, dt = \frac{\Gamma(p)}{a^{p}} \, \cos\left( \frac{p \pi}{2} \right) \end{align} then $I_{n}$ becomes \begin{align} I_{n} = \cos\left( \frac{\pi}{4n} \right) \, \Gamma\left( \frac{1}{2n} + 1 \right). \end{align}

For the case of $n=1$ the result is \begin{align} \int_{0}^{\infty} \cos(t^{2}) \, dt = \frac{1}{2} \sqrt{\frac{\pi}{2}}. \end{align}