What's wrong with this integral?

I'm trying to solve the integral:





sqrt(16 - x^2) / x





I keep coming up with the answer:





-4 * ln(abs((4 + sqrt(16 - x^2)/x)) + sqrt(16 - x^2) + C





But I'm told it's wrong and that the real answer is:





4 * ln((sqrt(16 - x^2) - 4) / abs(x)) + sqrt(16 - x^2) + C





The answers are VERY similar, and I can't tell if I'm just not simplifying all the way or if there is a problem in my work. Can anyone tell me why my answer is wrong? I've uploaded a copy of my work to here:





http://xfd.xanga.com/25ed850bd0c32139161...

What's wrong with this integral?
Your answer and the calculator answer are algebraically the same. ©





Edit: You can insert the negative from your answer: (I will concentrate only on the logarithmic expression.)





ln |x / {4+√(16-x²)}| ..... then rationalize (the factor is {4-√(16-x²)}


............... COMMENT: the expression given by the calculator is even in the reverse direction the radical is smaller than 4.





ln|x*{4-√(16-x²)} / [16 - (16-x²)]|


= ln|{4-√(16-x²)} / x| ....... note: there is an absolute value anyway. This is the answer given by the calculator.
Reply:The answers are in fact the same.


Change the -4ln(...) to 4ln(...)^(-1), so you get 4*ln(x / (4 + sqrt(16 - x^2))).


Multiply top and bottom by: 4 - sqrt(16 - x^2) and simplify. You will then find the answers are identical.

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