trouble solving the integral of $\cos(x^2)$

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No, I really mean the integral of $\cos(x^2)$, not $[\cos(x)]^2$. Can the chain rule be applied here?

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2 Answers

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The function $\cos(x^2)$ doesn't have an elementary primitive. It can't be expressed in terms of elementary functions, I mean. You can only study the definite integral using numerical methods.

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Represent $\cos x^{2}$ as a power series by $$ \cos x^{2}=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{4k}}{\left(2k\right)!}. $$ Then $$ \int\cos x^{2}dx=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}\int x^{4k}dx}{\left(2k\right)!}=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}x^{4k+1}}{\left(2k\right)!\left(4k+1\right)}+C $$ This is called the Fresnel integral.

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