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

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SUMMARY

The integral of 1/x is defined as ln(x), but a common point of confusion arises regarding the absence of absolute value signs in the argument of the logarithm. In the context of an ordinary differential equation (ODE) problem, the variable u = cos(t) can yield negative values. However, since the problem specifies an initial value condition where t is constrained to a range that keeps the argument of ln(x) positive, the absolute value can be disregarded. This clarification is crucial for understanding the application of logarithmic functions in calculus.

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  • Familiarity with ordinary differential equations (ODEs).
  • Knowledge of the properties of logarithmic functions.
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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.
 
Last edited:
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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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