Integrating the Square Root of a Fraction: How to Solve This Tricky Integral

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SUMMARY

The integral of the square root of a fraction, specifically \(\int \sqrt{\frac{1+x}{1-x}} dx\), can be solved using the substitution method and trigonometric identities. The final solution is \(\arcsin{x} - \sqrt{1-x^2} + C\). To simplify the integrand, multiplying by \(\frac{\sqrt{1+x}}{\sqrt{1+x}}\) and recognizing that \((1+x)(1-x) = 1 - x^2\) is crucial. Understanding the derivative of \(\arcsin(x)\) aids in approaching the integral effectively.

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Homework Statement



\int \sqrt {\frac{1+x}{1-x}} dx = ?

The solution is:

= \arcsin{x} -\sqrt{1-x^2} + const

If someone could help me to find the solution I'll be very pleased! :-)
Thanks everybody!
 
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Try multiplying the integrand by \frac{\sqrt{1+x}}{\sqrt{1+x}} and remember that (1+x)(1-x) = 1 - x^2. If you consider what the derivative of arcsin(x) is, it's easier to see what you should be trying to do to the integral. You may also need to do a substitution.
 

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