Integrating Sin(x^3) - Homework Equations & Solution

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Homework Statement


I need to intergrate sin(x^3) for a sum and I don't know how to.


Homework Equations


The sum is (integrate)3x+Sin(x^3)+1


The Attempt at a Solution


I've tried substituting u for x^3 but I don't know where to go from there considering du=3x^2.dx, which isn't relevant to my equation in any way.
 
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You are not going to find any elementary function anti-derivative for this.
 
That's what I figured. So what do I do with it?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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