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

May 21, 2016

You could use two ways - pure algebra or partial fractions - either of which give you $x - 10 \ln \left\mid x + 10 \right\mid + C$ as the final answer.

#### Explanation:

Algebraic Approach

First, realize that we can rewrite the integral as:
$\int \frac{x + 10 - 10}{x + 10} \mathrm{dx}$

Now we can split it up into two fractions, like so:
$\int \frac{x + 10}{x + 10} - \frac{10}{x + 10} \mathrm{dx}$
$= \int 1 - \frac{10}{x + 10} \mathrm{dx}$

Using the sum rule for integrals, this further simplifies to:
$\int 1 \mathrm{dx} - \int \frac{10}{x + 10} \mathrm{dx}$
$= \int 1 \mathrm{dx} - 10 \int \frac{1}{x + 10} \mathrm{dx}$

Evaluating these is pretty straightforward now:
$x + {C}_{1} - 10 \ln \left\mid x + 10 \right\mid + {C}_{2}$

Since ${C}_{1} + {C}_{2}$ is just another constant, we can lump them together in one general constant $C$:
$\int \frac{x}{x + 10} \mathrm{dx} = x - 10 \ln \left\mid x + 10 \right\mid + C$

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 $\frac{x}{x + 10}$ has only linear factors, we know the partial fraction decomposition will be something like:
$A + \frac{B}{x + 10}$

Setting it up, we have:
$\frac{x}{x + 10} = A + \frac{B}{x + 10}$

Multiplying through by $x + 10$ gives us:
$x = A \left(x + 10\right) + B$

If we let $x = - 10$, we can find the value of $B$:
$x = A \left(x + 10\right) + B$
$- 10 = A \left(- 10 + 10\right) + B$
$- 10 = B$

Now we have:
$x = A \left(x + 10\right) - 10$

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

Therefore $\frac{x}{x + 10} = 1 - \frac{10}{x + 10}$. Putting this back into the integral:
$\int 1 - \frac{10}{x + 10} \mathrm{dx}$
$= x - 10 \ln \left\mid x + 10 \right\mid + C \to$ as we discovered previously

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