mental blank (integration) (1 Viewer)

[slyball]

(ashamed to ask, but i just can't remember.....)

when integrating a logarithmic function, you usually end up with a number over number ouside of ln (bottom function). now i don't know where that bottom number in the number over number comes from.
for example,
question asking to integrate

x + 1 / x + 2x - 5

answer is... ln (x + 2x - 5) + C.

now, i don't know where the 2 in the comes from. the 1 comes from the differential of the top, yea? as for the 2, does that come from the coefficient of the differential of the bottom's x with the power (x) or the coefficient (or differential here..) of the bottom x with a power (2x)?

please help. i'm pullin my hair out over this.
thanks in advance!

 

[scandal]

to integrate log functions you need to have the derivative of a function over the function itself. Therefore for the function x2 +2x-5 the derivative is 2x + 2. But in the question you are given x + 1/x2 + 2x - 5. The link between the x + 1 and 2x + 2 is that x + 1 is half of 2x + 2. You can manipulate equations to look how you want them to look as long as they are still the same.

Therefore to make 2x + 2/x2 + 2x -5 look like the original equation you need to multiply by a half:

1/2 (2x + 2/x2 + 2x - 5) - now in the form of derivative of a function over a function therefore can be integrated -

1/2 ln(denominator) +c

= 1/2 ln(x2 + 2x - 5)

Hope that helps - check out some textbooks im sure they would have mroe worked examples