Determine the antiderivative of 1/(x+5)

To find the antiderivative of
1/(x+5).

Let f(x) =
1/(x+5).

To find the function F(x)  such that F'(x) = f(x)
= 1/(x+5).

We know that d/dx { log (ax +b)} =
(ax+b)'/(ax+b).

d/dx {log(ax+b)} =
a/(ax+b).

Therefore equating a/(ax+b) = 1/(x+5) we
get:

a(x+5) = ax+b

ax+5a =
ax+b

ax = ax.

5a = b . Or b =
5a.

Therefore ax+b =
ax+5a.

F(x) = log (ax+5a) = log
a(x+5).

Or F(x) = log(x+5)
+loga.

Therefore the anti derivative of 1/(x+5 ) is log
(x+5) + a constant.