How to calculate the value of 'n'? ∫dx/√(2ax - x² ) = a^n sin^-1[(x/a) -1]

Indranil

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How to calculate the value of 'n'? ∫dx/√(2ax - x² ) = a^n sin^-1[(x/a) -1]

∫dx/√(2ax - x² ) = a^n sin^-1[(x/a) -1] I don't understand the equation. Could you simplify the equation here and tell me how to calculate the value of 'n'?
 

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∫dx/√(2ax - x² ) = a^n sin^-1[(x/a) -1] I don't understand the equation. Could you simplify the equation here and tell me how to calculate the value of 'n'?
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First thing to do would be to calculate to calculate the anti-derivative of the function at the LHS (left-hand-side). Then compare the result with RHS.
 
∫dx/√(2ax - x² ) = a^n sin^-1[(x/a) -1] I don't understand the equation. Could you simplify the equation here and tell me how to calculate the value of 'n'?

It says the problem can be solved using dimensional analysis. Let's try it.

Suppose x is measured in meters. What must the units of a be? What units will be appropriate for the integral?

Then what will be the units of the right side? That will tell you what n has to be, if this is to be true.

Then, if you don't trust them, you could differentiate the right side to see that it gives the integrand. That will save a little work integrating.
 
It says the problem can be solved using dimensional analysis. Let's try it.

Suppose x is measured in meters. What must the units of a be? What units will be appropriate for the integral?

Then what will be the units of the right side? That will tell you what n has to be, if this is to be true.

Then, if you don't trust them, you could differentiate the right side to see that it gives the integrand. That will save a little work integrating.
I know a little bit of integration and differentiation but
I don't know how to integrate or differentiate these types of equations. Could you help me to do these types of integration and differentiation?
 
I know a little bit of integration and differentiation but
I don't know how to integrate or differentiate these types of equations. Could you help me to do these types of integration and differentiation?

Just do what I suggested, which doesn't require either integration or differentiation.

If you had to integrate this, you would complete the square and use a trig substitution; look that up in your textbook to learn more.

To differentiate the right-hand side, look up the derivative of the inverse sine, and use the chain rule.

But it sounds like this assignment was given without asking you to do either, since you haven't been taught how. (Or were you taught, but you didn't learn?)
 
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