Answer to Question #331099 in Calculus for ash

Question #331099

If c>0, then the following equality holds,


the integral from a to b dx/x equals to the integral from ac to bc dx/x.


Show your work and an explanation for each step.


1
Expert's answer
2022-04-20T14:05:57-0400

"\\displaystyle\\int_a^b \\frac{dx}{x}=\\int _{ac}^{bc}\\frac{dx}{x}" ;

let's integrate the left and right parts of the equality:

"\\displaystyle\\ln |x||_a^b =\\displaystyle\\ln |x||_{ac}^{bc}" ;

"\\displaystyle\\ln |b|-\\ln|a| =\\displaystyle\\ln |bc|-\\ln|ac|" ;

according to the property of the logarithm we can write the following formula:

"\\displaystyle\\ln\\left|\\frac ba\\right|=\\ln\\left|\\frac{bc}{ac}\\right|" ;

"\\displaystyle\\ln\\left|\\frac ba\\right|=\\ln\\left|\\frac ba\\right|" - the identity is proved.


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