# Simple integrals for gravitational potential

• MikeGomez
In summary, the conversation is about solving integrals using substitution, specifically \int \frac{dx}{(a+x)^2}. The solution involves substituting u = x+a and using the formula \int {(u)^{-2}}{du} = -\frac{1}{u} along with a constant c.
MikeGomez

## Homework Equations

I need help solving intergral…
$$\int \frac{dx}{(a+x)^2}$$

## The Attempt at a Solution

I found the integral for…
$$\int \frac{dx}{(a^2+x^2)}$$ = 1/a arctan x/a

But I don’t know how to apply that to the original integral which is a little different

$$\int \frac{dx}{(a+x)^2} = \int \frac{dx}{(a^2+x^2+2ax)}$$

I also need to solve the following integral
$$\int \frac{dx}{(a+b-x)^2}$$

It’s not homework. The reason is that I want to work through the numbers that Rybczyk gives as equation 1 for gravitational potential in his paper “Gravitational Effect on Light Propagation”

http://www.mrelativity.net/Gravitat...ravitational Effects on Light Propagation.htm

I have simplified his equation in my post somewhat, as I already know how to separate the two terms separated by the minus sign, and assuming the gravitational constant G and the mass of the bodies are constant, I know that they can come out in front of the integral sign.

Also, I changed the variable names to more familiar ones. I hope the variable name substitutions helps rather than hinders. If not I'll have to rewrite this whole post using Rybczyk's exact variables..

Thanks.

MikeGomez said:

## Homework Equations

I need help solving intergral…
$$\int \frac{dx}{(a+x)^2}$$

## The Attempt at a Solution

I found the integral for…
$$\int \frac{dx}{(a^2+x^2)}$$ = 1/a arctan x/a

But I don’t know how to apply that to the original integral which is a little different

$$\int \frac{dx}{(a+x)^2} = \int \frac{dx}{(a^2+x^2+2ax)}$$

I also need to solve the following integral
$$\int \frac{dx}{(a+b-x)^2}$$

It’s not homework. The reason is that I want to work through the numbers that Rybczyk gives as equation 1 for gravitational potential in his paper “Gravitational Effect on Light Propagation”

http://www.mrelativity.net/Gravitat...ravitational Effects on Light Propagation.htm

I have simplified his equation in my post somewhat, as I already know how to separate the two terms separated by the minus sign, and assuming the gravitational constant G and the mass of the bodies are constant, I know that they can come out in front of the integral sign.

Also, I changed the variable names to more familiar ones. I hope the variable name substitutions helps rather than hinders. If not I'll have to rewrite this whole post using Rybczyk's exact variables..

Thanks.
For the initial integral, use substitution, letting u = x+a .

Thanks Sammy. It looks to me like du = dx in this case. Is that right?

$$\int \frac{dx}{(a+x)^2} =$$
$$\int {(a+x)^{-2}}{dx}$$

u = a + x
du = dx

$$\int {(u)^{-2}}{du}=$$
$$-\frac{1}{u} + c$$
 made correction (forgot the + c)

Substituting a + x for u gives

$$-\frac{1}{a+x} + c$$

Is that right?

Last edited:
MikeGomez said:
Thanks Sammy. It looks to me like du = dx in this case. Is that right?

$$\int \frac{dx}{(a+x)^2} =$$
$$\int {(a+x)^{-2}}{dx}$$

u = a + x
du = dx

$$\int {(u)^{-2}}{du}=$$
$$-\frac{1}{u}$$
Substituting a + x for u gives
$$-\frac{1}{a+x}$$

Is that right?

Yes, it's right.

Will try same for the second part of original question.

NOTE: I editied post #3 to make a correction for the missing ( + c) for a constant.

## What is gravitational potential?

Gravitational potential is a measure of the gravitational potential energy per unit mass at a specific point in a gravitational field. It is a scalar quantity, meaning it has only magnitude and no direction.

## How is gravitational potential calculated?

The gravitational potential at a point is calculated by dividing the gravitational potential energy by the mass of the object. The formula for gravitational potential is V = -GM/r, where G is the gravitational constant, M is the mass of the object creating the gravitational field, and r is the distance from the object to the point where the potential is being measured.

## What is a simple integral for gravitational potential?

A simple integral for gravitational potential is an integral that can be easily solved using basic calculus techniques. In the case of gravitational potential, the integral is usually a single variable integral, with the variable being the distance from the object creating the gravitational field.

## Why is calculating gravitational potential important?

Calculating gravitational potential is important in understanding the behavior of objects in a gravitational field. It allows us to predict the motion and interactions of celestial bodies, as well as understand the structure and evolution of the universe.

## What is the difference between gravitational potential and gravitational potential energy?

Gravitational potential is a measure of the potential energy per unit mass at a specific point in a gravitational field, while gravitational potential energy is the total potential energy of an object in a gravitational field. Gravitational potential is a scalar quantity, while gravitational potential energy is a vector quantity, as it takes into account both magnitude and direction.

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