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So using the Inside Interesting Integrals book, I learned how to handle $\int^\infty_0\frac{\sin{x^2}}{x^2}dx$ & $\int^\infty_0\frac{\cos{x^2}}{x^2}dx$ (the latter doesn't converge naturally), and $\int^\infty_0\frac{\cos{x^2}}{\sqrt{x}}dx$ & $\int^\infty_0\frac{\sin{x^2}}{\sqrt{x}}dx$

But don't know how to evaluate $$\int^{\frac{\pi}{4}}_0\frac{\tan{x^2}}{x^2}dx$$ and $$\int^{\frac{\pi}{4}}_0\frac{\tan{x^2}}{\sqrt{x}}dx$$

Any techniques to tackle these integrals?

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    It doesn't seem that you can do anything about them, except for numerical evaluation. – 2017-01-01
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    I agree with Momo: Mathematica suggests no closed form for either of them, and WolframAlpha doesn't recognise decimal approximations of them. – 2017-01-01
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    hopeless...with some luck you can find a hypergeometric rep. here but nothing more – 2017-01-01
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    I'm not even able to find a Hyp.Rep. I'm just browsing through G&R – 2017-01-01
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    The upper limit seems random – 2018-04-20

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