How would one integrate $\sin\left(\pi t^2\right)$ from 0 to 1? I got this problem on my final exam and was completely lost on how to solve it. It's pretty short as problems go but it was surprisingly difficult. Does u-sub work here?
$$ \int_{0}^{1}\sin(\pi t^2) dt $$
Thanks for your help!

EDIT:
I'd appreciate if a mod could delete this. I appear to have remembered the problem incorrectly and it would be impossible to solve using things learned in Calc 1.
 A: Since you're in the range $[0,1]$ you can deal with Taylor Series
$$\sin(\pi t^2) = \sum_{k = 0}^{+\infty} \frac{(-1)^k (\pi t^2)^{2k+1}}{(2k+1)!}$$
Hence
$$\int_0^1 \sum_{k = 0}^{+\infty} \frac{(-1)^k (\pi t^2)^{2k+1}}{(2k+1)!} = \sum_{k = 0}^{+\infty} \frac{(-1)^k \pi^{2k+1}}{(2k+1)!}\int_0^1 t^{2(2k+1)}\ \text{d}t$$
The integral is trivial and you get at the end
$$\sum_{k = 0}^{+\infty} \frac{(-1)^k \pi^{2k+1}}{(2k+1)!} \frac{1}{4k+3}$$
If you are in search of numerical approximation then you can develop the series term by term:
$$\frac{\pi}{3} - \frac{\pi^3}{42} + \frac{\pi^5}{1212} - \frac{\pi^7}{10900} + \ldots $$
Numerical result:
$$0.5048545941136863...$$
A: This is likely asking for a numerical approximation or you didn't write it correctly. The exact answer is $\frac{\operatorname{S}(\sqrt{2})}{\sqrt{2}}$ where $\operatorname{S}(x)$ is the Sine Integral function, which is cannot be represented in terms of elementary functions (the basic functions you'll see in a calculus class). A numerical approximation is $0.504855...$
A: I believe you are looking at Fresnel's integral, particularly,
$$S(x)=\int_0^x\sin(t^2)\ dt$$
