https://i.stack.imgur.com/kT1M0.png
I'm really lost as to what to do. Do I need to Taylor expand around $\alpha=0$? And if so, how does the third expansion term even look like? I found the first and second terms to:
$$J_0=\int_{x_1}^{x_2}f(y,y_x,x)dx$$
$$J_1=\int_{x_1}^{x_2}\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial y_x}\frac{\partial y_x}{\partial \alpha}dx$$
Any help is appreciated!