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## integral of sin(x^2)

So you resurrected this thread from over a year ago just to say you did not understand it?

The original question was to integrate $sin(x^2)$, NOT $sin^2(x)$ for which your solution would be appropriate.

That was said back in November of 2009.
 I think I have a solution. I hope it was not so late :). tan(x^2)=m dx=cos(x^2)dm integral [sin(x^2)] = integral [mdm/(m^2+1)] m^2+1=a and 2mdm=da ........ integral [sin(x^2)] = 0.5*ln[(tan(x^2))^2+1]+c

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 Quote by Mstf_akkoc dx=cos(x^2)dm
Why is this true?

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 Quote by micromass Why is this true?
Well, if dx = cos(x^2) dm, then sec(x^2) = dm/dx and m=... I got lost.
 Actualy is not true. I have done a misteke when calculete the [tan(x^2)]' 2xdx=cos^2(x^2)dm is true if I find a solution with this I ll write.

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 Quote by Mstf_akkoc Actualy is not true. I have done a misteke when calculete the [tan(x^2)]' 2xdx=cos(x^2)dm is true if I find a solution with this I ll write.
No no no.

If tan(x^2) = m, then $$2x sec^2(x^2) dx = dm$$ and $$2x dx = cos^2(x^2) dm$$

Can you see why?

 Quote by HallsofIvy Why have you suddenly jumped into this thread from back in February? Did you not read the previous posts? As said in the very first response, $sin(x^2)$ does NOT have an elementary anti-derivative. After EquinoX told us that the problem was really $$\int_{y= 0}^5\int_{x= y^2}^{25} y sin(x^2)dx dy$$ it was suggested that he reverse the order of integration. Doing that it becomes $$\int_{x= 0}^{25}\int_{y= 0}^{\sqrt{x}} y sin(x^2)dy dx$$ $$= \frac{1}{2}\int_{x= 0}^{25}\left[y^2\right]_{y=0}^{\sqrt{x}} sin(x^2) dx$$ $$= \frac{1}{2}\int_{x= 0}^{25} x sin(x^2) dx$$ which can be integrated by using the substitution $u= x^2$: If $u= x^2$, du= 2x dx so x dx= (1/2)du. When x= 0, u= 0 and when x= 25, u= 625 so the integral is $$\frac{1}{4}\int_0^{625} sin(u) du= -\frac{1}{4}\left[cos(u)\right]_0^{625}$$ $$= -\frac{1}{4}(-0.984387)= 0.246097$$
I have a problem. Isn't:
$$\left[cos(u)\right]_0^{625} = cos(625)-cos(0) = (-0.984387)-(1)=-1.984387$$

Therefore;
$$\frac{1}{4}\int_0^{625} sin(u) du = -\frac{1}{4}(-1.984387)= 0.496097$$