Are These Formulas for Indefinite Integrals and Antiderivatives Correct?

[Jhenrique]

$$\int \frac{d}{dx}f(x)dx = f(x) + C_x$$ $$\iint \frac{d^2}{dx^2}f(x)dx^2 = f(x) + xC_x + C_{xx}$$
$$\int \frac{\partial}{\partial x}f(x,y)dx = f(x,y) + g_x(y)$$ $$\int \frac{\partial}{\partial y}f(x,y)dy = f(x,y) + g_y(x)$$
$$\iint \frac{\partial^2}{\partial x^2}f(x,y)dx^2 = f(x,y) + x g_{x}(y) + g_{xx}(y)$$ $$\iint \frac{\partial^2}{\partial x \partial y}f(x,y)dxdy = f(x,y) + \int_{y_0}^{y}g_x(y)dy + G_x(y_0) + g_y(x)$$ $$\iint \frac{\partial^2}{\partial y \partial x}f(x,y)dydx = f(x,y) + \int_{x_0}^{x}g_y(x)dx + G_y(x_0) + g_x(y)$$ $$\iint \frac{\partial^2}{\partial y^2}f(x,y)dy^2 = f(x,y) + y g_y(x) + g_{yy}(x)$$

I was trying apply the idea of indefinite integral (ie, the antiderivative of a function + a arbitrary constant/function) for all possible cases. You think that all equation above are correct?

 

[HallsofIvy]

I would NOT write "$dx^2$" for a double integral. And, frankly, I can see no point in writing out all those integrals!

 

[Jhenrique]

IMO, those integral are usefull for didactic efect.

 

[Mark44]

I would NOT write "$dx^2$" for a double integral. And, frankly, I can see no point in writing out all those integrals!

+1 to both.

IMO, those integral are usefull for didactic efect.

For what purpose - to be memorized?

 

[micromass]

IMO, those integral are usefull for didactic efect.

In my mathematical career, I have never needed integral tables like in the OP.

 

[micromass]

Anyway, you want to know whether the formulas are correct? Well, then give us a proof of the formula and we'll tell you if the proof is right or wrong.

 

[Jhenrique]

In my mathematical career, I have never needed integral tables like in the OP.

How you express a family of antiderivative of a function (f'(x)) that haven't integral?

$\int f'(x) dx = \int_{x_0}^{x} f'(x) dx + f(x_0) + C$

But if you want to integrate twice a function (f''(x) (in this case)) that haven't integral in terms of elementary functions? Answer:

$\iint f''(x) dxdx = \iint_{x_0 x_0}^{x\;x} f''(x) dx dx + f'(x_0)(x - x_0) + f(x_0) + xC_1 + C_2$

Why Riemann did want to study and develop his theory of differential geometry if hadn't application? Why he wanted. Sometimes a theory haven't practice application but has just theoretical application.

 

[Mark44]

How you express a family of antiderivative of a function (f'(x)) that haven't integral?

$\int f'(x) dx = \int_{x_0}^{x} f'(x) dx + f(x_0) + C$

But if you want to integrate twice a function (f''(x) (in this case)) that haven't integral in terms of elementary functions? Answer:

$\iint f''(x) dxdx = \iint_{x_0 x_0}^{x\;x} f''(x) dx dx + f'(x_0)(x - x_0) + f(x_0) + xC_1 + C_2$

What good does having this formula do for you? If f'' doesn't have an antiderivative in terms of elementary functions, then how are you going to get f', or for that matter, f?

Why Riemann did want to study and develop his theory of differential geometry if hadn't application? Why he wanted. Sometimes a theory haven't practice application but has just theoretical application.