Question

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.

Answer 1

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

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.