# Tagged Questions

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### Evaluating $\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$

$$\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$$ How to solve this trigonometric integral? I can't find any solutions. Some books suggest to use Fresnel integral. I would be grateful if you could ...
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### Need help with $\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx$

Let $$I=\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx,\tag1$$ where $$S(x)=-\frac12+\int_0^x\sin\frac{\pi t^2}2dt,\tag2$$ ...
### how to prove an identity related to $\int_0^\infty\sin(x^{1+a})dx$?
i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$ and the computer give me the following result ...
### Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts
$$\int_0^1 \! C(x) \, \mathrm{d} x.$$ where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$. I am really not quite sure how to go about this one, especially given that it needs to be calculated ...