Consider the second of these integrals (What's the meaning of second right here?)
$$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{x}}\frac{\partial \dot{x}}{\partial \alpha})\mathrm dx$$ Consider the second of these integrals: $$\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx=\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x \partial \alpha}\mathrm dx$$
What did they mean by "second"? There must be a negative on LHS or RHS. What happened to the line? To me, it seems like he had used chain rule for $\partial y$. But, why there's no negative?
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In the post, actually $\dot{y}=\frac{dy}{dx}$. So,
$$\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx=\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x \partial \alpha}\mathrm dx$$
In the equation, they just wrote $\frac{dy}{dx}$ instead of $\dot{y}$
What did they mean by "second"?
They've mentally expanded $$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}\left(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\right)\mathrm dx$$ (which I've corrected to be what it says in the image rather than your MathJax) as $$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha} \mathrm dx + \int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx$$
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