what are the steps to integrate the following without using numerical methods-
$$ \int \sin(x^2)\,dx.$$
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3This is a special function : a Fresnel integral see too this link. – Raymond Manzoni Jul 07 '16 at 09:30
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1The antiderivative of $sin(x^2)$ is not an elementary function. – Chris Sanders Jul 07 '16 at 09:31
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2You may be interested in this. – David Mitra Jul 07 '16 at 09:32
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2Matthew Wiener proved it non elementary here. – Raymond Manzoni Jul 07 '16 at 09:37
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1If 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? – J. M. ain't a mathematician Jul 07 '16 at 10:16
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1You might also have a look at this post and other questions linked there. However, they are mainly about $\int_0^\infty \sin(x^2) dx$. – Martin Sleziak Jul 07 '16 at 11:05
1 Answers
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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:
$$\int a^b\space\text{d}a=\frac{a^{b+1}}{b+1}+\text{C}$$
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(1+2n)!}\int x^{2+4n}\space\text{d}x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{3+4n}}{(4n+3)(1+2n)!}+\text{C}$$
Jan Eerland
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