# How do you integrate f(x) = (x+6)/(x+1)?

Feb 19, 2015

we can rewrite the function into a form which is easily integrated

Think of the function like this

$\frac{x + 6}{\left(1\right) \left(x + 1\right)} = \frac{A}{1} + \frac{B}{x + 1}$

We are doing a partial fraction decomposition.
Multiply the expression above by $\left(1\right) \left(x + 1\right)$

$x + 6 = A \left(x + 1\right) + B \left(1\right)$

$x + 6 = A x + A + B$

Equating coefficients we get

$A = 1$ and $A + B = 6$

Since $A = 1$ we can conclude that $B = 5$

Therefore we can rewrite as follows

$\int \frac{x + 6}{x + 1} \mathrm{dx} = \int 1 + \frac{5}{x + 1} \mathrm{dx}$

Integrating we get

$x + 5 \ln | x + 1 | + C$