Question

How do you integrate `(x)/(x+10) dx`?

Answer 1

You could use two ways - pure algebra or partial fractions - either of which give you `x-10lnabs(x+10)+C` as the final answer.

Explanation:

Algebraic Approach

First, realize that we can rewrite the integral as:
`int(x+10-10)/(x+10)dx`

Now we can split it up into two fractions, like so:
`int(x+10)/(x+10)-10/(x+10)dx`
`=int1-10/(x+10)dx`

Using the sum rule for integrals, this further simplifies to:
`int1dx-int10/(x+10)dx`
`=int1dx-10int1/(x+10)dx`

Evaluating these is pretty straightforward now:
`x+C_1-10lnabs(x+10)+C_2`

Since `C_1+C_2` is just another constant, we can lump them together in one general constant `C`:
`intx/(x+10)dx = x-10lnabs(x+10)+C`

Partial Fractions Approach

Alternatively, if we want some practice with partial fractions or the teacher is forcing us to use this method, we can do it a little differently.

Since our original fraction `x/(x+10)` has only linear factors, we know the partial fraction decomposition will be something like:
`A+B/(x+10)`

Setting it up, we have:
`x/(x+10)=A+B/(x+10)`

Multiplying through by `x+10` gives us:
`x=A(x+10)+B`

If we let `x=-10`, we can find the value of `B`:
`x=A(x+10)+B`
`-10=A(-10+10)+B`
`-10=B`

Now we have:
`x=A(x+10)-10`

We can let `x` equal anything now to find `A`. For simplicity, let's have `x=0`:
`0=A(0+10)-10`
`10=10A`
`A=1`

Therefore `x/(x+10)=1-10/(x+10)`. Putting this back into the integral:
`int1-10/(x+10)dx`
`=x-10lnabs(x+10)+C->` as we discovered previously

I would not suggest the partial fractions method unless you were required to use it.