You can't express this integral in terms of elementary functions. This integral is known as the Fresnel integral. You can express the integral as a power series instead: \(\displaystyle S(x) = \int_0^x \sin(t^2) \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+3}}{(2n+1)!(4n+3)} \).

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TopNewest@Dollesin Joseph Use this result: \(\displaystyle \int\limits_0^{\infty} \sin(x^n) dx = \sin(\frac{\pi}{2n})\frac{\Gamma(n+1)}{\Gamma(n)}\)..

that is only for these limits... other wise mention limits

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You can't express this integral in terms of elementary functions. This integral is known as the Fresnel integral. You can express the integral as a power series instead: \(\displaystyle S(x) = \int_0^x \sin(t^2) \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+3}}{(2n+1)!(4n+3)} \).

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