What is the indefinite integral of the function f(x)=1/(x^2+8x+16)?

We notice that the denominator is a perfect square: x^2 +
8x + 16 = (x+4)^2

We'll re-write the
integral:

Int f(x)dx = Int
dx/(x+4)^2

We'll apply the techinque of substitution of the
variable.

We'll replace x+4 by
t.

x+4 = t

We'll differentiate
both sides:

(x+4)'dx = dt => dx =
dt

We'll re-write the integral in the variable
t:

Int dx/(x+4)^2 = Int
dt/t^2

Int dt/t^2 = Int
[t^(-2)]*dt

Int [t^(-2)]*dt = t^(-2+1)/(-2+1) +
C

Int [t^(-2)]*dt = t^(-1)/-1 +
C

Int [t^(-2)]*dt = -1/t +
C

But t =
x+4

The requested indefinite integral of the
function is: Int dx/(x^2 + 8x + 16) = -1/(x+4) + C