Question

integrate abs(log(5 - x/2)) from 0 to 9, how would you start with this. The base of the log is 10, not e

Answer 1

\[\large \int_0^9\left|\log\left(5-\frac{x}{2}\right) \right|\, dx \] Since you have an absolute value, you need to consider how to break up the expression for the values for which the integrand is negative or positive for certain values of x:

By definition of the absolute value:
\[\large \left|\log\left( 5-\frac{x}{2}\right) \right| = \begin{cases} \log\left( 5-\frac{x}{2}\right) , & \text{if} \log\left( 5-\frac{x}{2}\right) \ge 0 \\ -\log\left( 5-\frac{x}{2}\right) , & \text{if} \log\left( 5-\frac{x}{2}\right) < 0\end{cases} \]
So solve for x in each case to determine when your integrand is positive or negative.
Case 1:
\[ \log\left( 5-\frac{x}{2}\right) \ge 0\\
5-\frac{x}{2}\ge10^0\\
-\frac{x}{2}\ge-4\\
x \le 8\], so consider the positive case when x is less or equal to 8 (I.e. for the interval 0 to 8)

Case 2:
\[\log\left( 5-\frac{x}{2}\right) < 0\\
5 - \frac{x}{2}<10^0\\
-\frac{x}{2}<-4\\
x>8\], so use this branch for values greater than 8, i.e. the interval 8-9
(technically , 8 < x< 10 because after x=10, the logarithm has a negative argument, but we don't need to worry about this technicality.. it's just a restriction of the domain).

Thus, you need to find the integral:
\[\large \int_0^9\left|\log\left(5-\frac{x}{2}\right) \right|\, dx =\int_0^8\log\left(5-\frac{x}{2}\right)\, dx+\int_8^9-\log\left(5-\frac{x}{2}\right)\, dx\]

Answer 2

whatever got cut off at the end is just a "dx"

Answer 3

ahh it makes sense, I know more than basic integrations but I've never come across to integrate a log that isnt natural. First thought that comes to my mind is doing a integration by parts. Is that alright?

Answer 4

Making u=log(5-x/2) and dv=dx

Answer 5

Yes, I would go with integration by parts. That is also how they get the integral of ln x. The process should be similar :)

Although it might be a bit less cumbersome (depending on your view...) to do a u-substitution first u = 5-x/2, -2du = dx
And then apply integration by parts on that result. It should come out to the same :)

Answer 6

you are right, much less numbers. Thank you very much for this instruction!

Answer 7

your welcome!