How Can I Solve This Improper Integral With Absolute Value?

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SUMMARY

The discussion focuses on evaluating the improper integral of the function 1/sqrt(|1-4x^2|) from 0 to 1. The correct approach involves splitting the integral into two parts: from 0 to 1/2 and from 1/2 to 1. This method ensures that the absolute value is properly addressed by applying trigonometric substitutions for each segment of the integral, specifically integrating (1/sqrt(1-4x^2)) from 0 to 1/2 and (1/sqrt(4x^2-1)) from 1/2 to 1.

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dan
Hi there
I'm having trouble solving this integral!

Evaluate the improper integral
int[1/(sqrt{absolute value(1-4x^2)})]dx where the upper limit=1, lower limit=0

without ignoring the absolute value sign?


Thank you in advance
 
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What exactly do you mean by "without ignoring the absolute value"?
Since this has an absolute value, one CAN'T do while ignoring the absolute value!

The obvious way to integrate this is to separate into two integrals, one from 0 to 1/2 and the other from 1/2 to 1:
integral (x=0 to 1/2) (1/sqrt(1- 4x^2))dx+ integral(x= 1/2 to 1)(1/sqrt(4x^2- 1))dx and then use trigonometric substitutions.

Using the definition of absolute value is certainly NOT ignoring it!
 

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