Understanding Integrals: Solving for dt in \frac{dr}{\sqrt{2\frac{GM}{r}+ 2C}

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In summary, solving for dt in the integral \frac{dr}{\sqrt{2\frac{GM}{r}+ 2C}} involves using substitution and integration by parts. First, substitute u = \sqrt{2\frac{GM}{r}+ 2C} and du = -\frac{GM}{r^2}dr to simplify the integral to \int\frac{dr}{u}. Then, use integration by parts with u = 1 and dv = \frac{dr}{u} to solve for dt. This process allows for the integration of complex functions and is essential in many areas of mathematics, physics, and engineering.
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PMP
What is the integral of:

[tex]dt= \frac{dr}{\sqrt{2\frac{GM}{r}+ 2C}}[/tex]​

where C is a constant?

I need to integrate this, but I don't know integrals so much. Thanks :rolleyes:
 
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  • #2
For simplicity, reduce your questions to general ones with as few constants as possible. (you can plug in whatever the specific constants are later). Are you asking how to compute:

[tex]\int \frac{dx}{\sqrt{1+a/x}} [/tex]?

If so, try the substitution [itex]u=\sqrt{1+a/x}[/tex].
 
  • #3
Yes, I am asking that integral. Thank you!
 

FAQ: Understanding Integrals: Solving for dt in \frac{dr}{\sqrt{2\frac{GM}{r}+ 2C}

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to solve problems involving continuous change, such as finding the distance traveled by an object over time or the amount of water flowing through a pipe.

What does "dt" represent in the integral formula?

"dt" represents the infinitesimal change in the independent variable, in this case time, as the integral is being solved. It is a crucial part of the integral formula as it allows for the calculation of the total change over a continuous period.

How do you solve for dt in the integral formula?

To solve for dt, you must first rearrange the formula to isolate it on one side. In the case of the given integral, you would divide both sides by the square root of the entire expression within the integral. This will leave dt on its own on the left side of the equation.

What is the significance of "G" and "M" in the integral formula?

"G" represents the gravitational constant, which is a fundamental constant in physics that relates the strength of the gravitational force to the masses and distances of objects. "M" represents the mass of the object that is causing the gravitational force. These values are used to calculate the gravitational potential energy in the integral formula.

How do you determine the value of "C" in the integral formula?

"C" is known as the constant of integration and can be determined by using initial conditions or boundary conditions specific to the problem being solved. In some cases, it may be given in the problem statement. It is essential to include this constant in the final solution to account for any potential variations or offsets in the initial conditions.

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