How can I compute this integral ∫ cos 2 ⁡ ( t x 2 − 1 ) d x

Question

How can I compute this integral  ∫      cos          2        ⁡      (    t                  x                  2                    −      1        )    d  x

Carter Escobar · Accepted Answer

Your integral is related with the Fourier cosine transform of

\frac{1}{\sqrt{1 + x}}

, that can be computed through Fresnel integrals, but is not an elementary function. For instance:

\begin{array}{rcl} \int_{1}^{M} \cos (t \sqrt{x^{2} - 1}) d x & = & \frac{1}{2} \int_{0}^{\sqrt{M^{2} - 1}} \frac{\cos (t u)}{\sqrt{u + 1}} d u \\ = & \frac{1}{2} \int_{1}^{1 + \sqrt{M^{2} - 1}} \frac{\cos (t u) \cos (t) + \sin (t u) \sin t}{\sqrt{u}} d u \\ = & \int_{1}^{\sqrt{1 + \sqrt{M^{2} - 1}}} \cos (t) \cos (t v^{2}) + \sin (t) \sin (t v^{2}) d v . \end{array}

Obviously,

\cos^{2} (z) = \frac{1 + \cos (2 z)}{2}

How can I compute this integral ∫ cos <mrow class="MJX-TeXAtom-

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