Question

integrating a logarithm

Answer 1

\[\int \frac{\text dx}{x}=\ln |x|+c=\ln |x|+\ln \left|e^c\right|=\ln |e^cx|=\ln |kx|\]
\[\int\limits_{x_1}^{x_2} \frac{\text dx}{x}=\left.\ln|x|~\right|_{x_1}^{x_2}=\ln |x_1|-\ln |x_2|=\ln \left|\frac{x_1}{x_2}\right|=\ln |x_1|-\ln |x_2|\]

Answer 2

i guess my question is

isn't the best place for the arbitrary constant of integration, {when taking the indefinite integral of a logarithm}, inside the logarithm ?

Answer 3

you are not taking the indefinite integral of a logarithm

Answer 4

also \[\int\limits_{x_1}^{x_2} \frac{\text dx}{x}=\left.\ln|x|~\right|_{x_1}^{x_2}=\ln |x_2|-\ln |x_1|\]

Answer 5

oh yeah , got those limits backwards in the definite integral .

but my question is about the indefinite integral

Answer 6

why do integration tables say
\[\int \frac{\text dx}{x}=\ln |x|+c\]
instead of \[\int \frac{\text dx}{x}=\ln |kx|\]

Answer 7

the latter is so much simpler

Answer 8

both seem really simple to me. the first one is how all the others are written

Answer 9

Not all integration tables list the constant of integration at all. The classic Gradshteyn & Ryshik do not, for example. You're just supposed to be mathematically sophisticated enough to realize the integral is only defined up to a constant.

So what would an integration table do which decided to save some printing space by leaving off the "+ C" after every indefinite integral do when listing the integral for 1/x? Have a constant in that integral alone? Leave off the k, and make the formula look different than other tables?