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I'm self teaching from Core Maths for Advanced Level by Bostock and Chandler. They say this:

$$\int \frac{1}{ax + b} dx = \frac{1}{a} ln |ax + b| + K = \frac{1}{a} ln A|ax + b|$$

There's no explanation of where the $A$ comes from, or where the constant of integration went. Can anyone explain how the equation is true, and what the motivation for doing this is?

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$A$ is where $K$ went. – Raskolnikov May 6 '12 at 10:12
up vote 1 down vote accepted

If $K = \frac1a \log_e A$, i.e. letting $A=e^{aK}$, then you should be able to combine the logs.

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It follows from the fact that $\ln uv=\ln u+\ln v$. In this case set $A=e^K$ and take $u=A$ and $v=|ax+b|$: then $\ln A|ax+b|=\ln A+\ln|ax+b|=\ln e^K+\ln|ax+b|=K+\ln|ax+b|$.

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$$\log (A |ax+b|) = \log A + \log |ax+b|.$$ Put $aK=\log A$.

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because we have deal with logarithms,i think author decided to use logarithms totaly, generaly $ln(a)+ln(b)=ln(a*b)$,so author had took $A=(1/a)*ln(k)$ ,because as k is constant,A is also constant,so no changing anything

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