Integral of 1/x = ln(x) Problem has missing absolute value?

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The discussion centers on the integral of 1/x equating to ln(x) and the confusion regarding the absence of absolute value signs in the solution. The user questions why the argument of ln() is not presented with absolute values, especially since u = cos(t) can yield negative values. The clarification provided states that since this is an initial value problem with a specific t, the argument of the natural log remains positive, allowing the omission of absolute values. This highlights the importance of context in calculus when dealing with logarithmic functions. Understanding the conditions of the problem is essential for correctly applying the integral.
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Integral of 1/x = ln(x)... Problem has missing absolute value!?

This is an ODE problem and solution, but what I'm not understanding here is calculus based:

2rwatya.jpg


Why does the argument to the red-boxed ln() not in absolute value signs?? ( | | )
u = cos(t) so the numerator can be negative. I am just confused as to why they left that out.

Thanks for any help.
 
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My solution is that, it is an Initial value problem, (there is only one t which we care about) and with that t the argument in the natural log is positive and so u can ignore the abs value!
 

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