Answered: how did you get -ln(A-T)=kt + C ? I'm…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 3TI: Find the average rate of change of f(x)=x2+2x8 on the interval [5,a] in simplest forms in terms ofa.

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how did you get -ln(A-T)=kt + C ? I'm not understanding where the negative came from...the integral of 1/A-T is ln(A-T) and the integral of k is kt...where is the negative coming from?

Expert Solution

Step 1: Doubt clearance for -ln(A-T)

Here the intergrand is

$fraction numerator 1 over denominator A minus T end fraction$ which is integrated with respect to T. Here the coefficient of T is (-1) in the denominator, so we multiply (-1) with ln(A-T) thereby getting -ln(A-T).

Another way.

Put U= A-T

dU=-dT

Substitute in $integral fraction numerator 1 over denominator A minus T end fraction space d T space equals space integral 1 over U left parenthesis negative d U right parenthesis space equals negative integral 1 over U d U equals negative ln space U equals negative ln left parenthesis A minus T right parenthesis$