Calculating $\int_{0}^{\infty}\sin(x^{2})dx$ I am supposed, in an  exercise, to calculate the above integral by integrating $f(z) = e^{-z^{2}}$ on the following countor: 
I began by separating the path $\gamma$ into three paths (obvious from the picture), and parametrizing each as follows:
$\gamma_{1} : [0, R] \rightarrow \mathbb{C}$ with $\gamma_{1}(t) = t$
$\gamma_{2} : [0, \frac{\pi}{4}] \rightarrow \mathbb{C}$ with $\gamma_{2}(t) = Re^{it}$
$\gamma_{3} : [0, \frac{\sqrt{2}R}{2}] \rightarrow \mathbb{C}$ with $\gamma_{3}^{-}(t) = t + it$ (with reverse orientation).
Then we can say that $\displaystyle\int_{\gamma} f(z) dz = \displaystyle\int_{\gamma_{1}} f(z) dz + \displaystyle\int_{\gamma_{2}} f(z) dz - \displaystyle\int_{\gamma_{3}^{-}} f(z) dz = 0$ since the path is closed.
Now $\displaystyle\int_{\gamma_{1}} f(z) dz = \displaystyle\int\limits_{0}^{R} e^{-t^{2}} dt$. We also get $\displaystyle\int_{\gamma_{3}^{-}} f(z) dz = -(i + 1) \displaystyle\int\limits_{0}^{\frac{\sqrt{2}R}{2}}e^{-2it^{2}} dt$. After playing around with sine and cosine a bunch to evaluate that last integral, I get:
$$0 = \int\limits_{0}^{R} e^{-t^{2}} dt + \int\limits_{\gamma_{2}} f(z) dz - \frac{i + 1}{\sqrt{2}} \int\limits_{0}^{R} \cos(u^{2}) du + \frac{i - 1}{\sqrt{2}} \int\limits_{0}^{R} \sin(u^{2}) du$$
I could not evaluate the integral along the second path, but I thought it might tend to 0 as $R \rightarrow \infty$. Then taking limits and equating real parts we get
$$\frac{\sqrt{2 \pi}}{2} = \displaystyle\int\limits_{0}^{\infty} \sin(u^{2}) du + \displaystyle\int\limits_{0}^{\infty} \cos(u^{2}) du$$
If I could argue that the integrals are equal, I would have my result.. But how do I?
So I need to justify two things: why the integral along $\gamma_{2}$ tends to zero and why are the last two integrals equal.

Your parametrisation of the third integral is rather complicated. Why not just write $\gamma_3:[0,R]\rightarrow \mathbb{C}$, $t\mapsto -e^{\pi i/4}t$. Then the integral becomes
$$
\int_R^0 e^{-e^{\pi i/2}t^2}e^{\pi i/4}dt = \int_R^0 e^{-it^2}e^{\pi i/4}dt =
e^{\pi i/4} \int_R^0 \cos t^2 - i \sin t^2 dt.
$$
I am sure you can take it from there.
As for bounding the integral $\int_{\gamma_2}e^{-z^2}dz$, the length of the contour grows linearly with $R$. How fast does the maximum of the integrand decay? It's the standard approach, using the fact that
$$
\left|\int_\gamma f(z) dz\right|\leq \sup\{|f(z)|: z \in \text{ image of }\gamma\}\cdot \text{length of }\gamma.
$$
