integrating to ln (2 Viewers)

[Treedom]

This might be a dumb question but evaluating

becomes this right

But if you factor out a quarter from the start to get

then it becomes

Now I'm pretty sure these answers aren't the same so I'm just very confused, is there a logarithm rule that explains this??

 

[InteGrand]

This might be a dumb question but evaluating

becomes this right

But if you factor out a quarter from the start to get

then it becomes

Now I'm pretty sure these answers aren't the same so I'm just very confused, is there a logarithm rule that explains this??

 

[pikachu975]

Basically this has to do with the +C you get when integrating.

For the first case, it becomes 1/4 * ln(4(x+1)) + C which then becomes 1/4 * (ln4+ln(x+1)) + C using log laws. Then from this, the C can be combined with the ln4 since ln4 is a constant, so C + 1/4 * ln4 will create a new constant, let's say D. Thus, 1/4 * ln(x+1) + D.

For the second case, 1/4 * ln(x+1) + C, here the C would be equal to the D of the first case.

SUMMARY: The ln4 gets absorbed by the constant to make a new constant.

 

[Treedom]

That clears a lot of confusion, thanks so much guys!