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what are the steps to integrate the following -
Not numerical method , but the integrated function is required.
$$ \int \sin(x^2)\,dx.$$

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closed as off-topic by user91500, Morgan Rodgers, Davide Giraudo, Watson, C. Falcon Jul 7 '16 at 14:53

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    $\begingroup$ This is a special function : a Fresnel integral see too this link. $\endgroup$ – Raymond Manzoni Jul 7 '16 at 9:30
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    $\begingroup$ The antiderivative of $sin(x^2)$ is not an elementary function. $\endgroup$ – Yon Teh Jul 7 '16 at 9:31
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    $\begingroup$ You may be interested in this. $\endgroup$ – David Mitra Jul 7 '16 at 9:32
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    $\begingroup$ Matthew Wiener proved it non elementary here. $\endgroup$ – Raymond Manzoni Jul 7 '16 at 9:37
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    $\begingroup$ If you're averse to using special functions like the Fresnel integral, then you must resort to numerical quadrature, and for evaluating the Fresnel integral numerically, you must use numerics anyway. But for symbolic purposes, the Fresnel result is fine. So, which? $\endgroup$ – J. M. is a poor mathematician Jul 7 '16 at 10:16
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Use:

$$\sin\left(x^2\right)=\sum_{n=0}^{\infty}\frac{(-1)^n\left(x^2\right)^{1+2n}}{(1+2n)!}$$

So, we get:

$$\int\sin\left(x^2\right)\space\text{d}x=\int\sum_{n=0}^{\infty}\frac{(-1)^n\left(x^2\right)^{1+2n}}{(1+2n)!}\space\text{d}x=\sum_{n=0}^{\infty}\frac{(-1)^n}{(1+2n)!}\int\left(x^2\right)^{1+2n}\space\text{d}x=$$


Now, use:

$$\int a^b\space\text{d}a=\frac{a^{b+1}}{b+1}+\text{C}$$


$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(1+2n)!}\int x^{2+4n}\space\text{d}x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{3+4n}}{(4n+3)(1+2n)!}+\text{C}$$

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