Integral of $x^2 \sin(x^2)$

Question

I was playing around learning the SymPy syntax (python library) e then I saw an example that intrigued me.

$$\int x^2\sin(x^2)dx = \frac{5x\cos(x^2)\Gamma(\frac{5}{2})}{8\Gamma({\frac{9}{4})}} + \frac{5\sqrt{2\pi}C(\frac{x\sqrt{2}}{\sqrt{\pi}})\Gamma{(\frac{5}{4})}}{16\Gamma{(\frac{9}{4})}}$$

Does anyone know which integration method (series expansions, transforms or whatever) is used to calculate it ? And what this $C$ means ?

Answer 1

With change of variable $X=x^2\quad;\quad dx=\frac{dX}{2\sqrt{X}}$ :

$$I=\int x^2\sin(x^2)dx=\int X\sin(X)\frac{dX}{2\sqrt{X}}=\frac12\int \sqrt{X}\sin(X)dX$$ Integration by part : $$I=-\frac12\sqrt{X}\cos(X)+\frac14\int\frac{\cos(X)}{\sqrt{X}}dX$$ $$I=-\frac12x\cos(x^2)+\frac14\int\frac{\cos(x^2)}{x}2xdx$$ $$I=-\frac12x\cos(x^2)+\frac12\int\cos(x^2)dx$$ $\int\cos(x^2)dx=\sqrt{\frac{\pi}{2}}\text{C}\left(\sqrt{\frac{2}{\pi}}\:x\right)+$constant

C$(t)$ is a special function called Fresnel Integral : http://mathworld.wolfram.com/FresnelIntegrals.html

In math softwares this function is commonly named FresnelC.

In the present case $t=\sqrt{\frac{2}{\pi}}\:x$ .

$$I=\int x^2\sin(x^2)dx=-\frac12x\cos(x^2)+\frac12\sqrt{\frac{\pi}{2}}\text{C}\left(\sqrt{\frac{2}{\pi}}\:x\right)+\text{constant}$$

This is exactly the same result than $$\int x^2\sin(x^2)dx = -\frac{5x\cos(x^2)\Gamma(\frac{5}{4})}{8\Gamma({\frac{9}{4})}} + \frac{5\sqrt{2\pi}C(\frac{x\sqrt{2}}{\sqrt{\pi}})\Gamma{(\frac{5}{4})}}{16\Gamma{(\frac{9}{4})}}+\text{constant}$$ because $\frac{\Gamma(\frac54)}{\Gamma(\frac94)}=\frac45$ .