Tweaked version of Fresnel's Integral

Calculus Level 3

It is known that

sin ( x 2 ) d x = cos ( x 2 ) d x = π 2 1.253314 \int_{-\infty}^{\infty} \sin(x^2)\,dx=\int_{-\infty}^{\infty} \cos(x^2)\,dx=\sqrt{\frac{\pi}2}\approx1.253314\dots

But what is the value of the following integral?

sin ( x 2 + 1 x 2 ) d x \int_{-\infty}^{\infty} \sin\left(x^2+\frac1{x^2}\right)\,dx


The answer is 0.618072606621.

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1 solution

Theo Cannon
Aug 1, 2018

sin ( x 2 + 1 x 2 ) d x = sin ( ( x 1 x ) 2 + 2 ) d x = sin ( x 2 + 2 ) d x By Glasser’s Master Theorem/Cauchy-Schl o ¨ milch transformation = cos ( 2 ) sin ( x 2 ) d x + sin ( 2 ) cos ( x 2 ) d x = ( cos ( 2 ) + sin ( 2 ) ) π 2 0.6180726 \begin{aligned} \int_{-\infty}^{\infty} \sin\left(x^2+\frac1{x^2}\right)\,dx&=\int_{-\infty}^{\infty} \sin\left(\left(x-\frac1x\right)^2+2\right)\,dx \\ &=\int_{-\infty}^{\infty} \sin\left(x^2+2\right)\,dx&\text{ By Glasser's Master Theorem/Cauchy-Schlömilch transformation } \\ &=\int_{-\infty}^{\infty} \cos(2)\sin(x^2)\,dx+\int_{-\infty}^{\infty} \sin(2)\cos(x^2)\,dx \\ &= (\cos(2)+\sin(2))\sqrt{\frac{\pi}2}\\&\approx0.6180726\dots\end{aligned}

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