Find the indefinite integral of y=1/(x^2+4x+4)

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We'll write the denominator as the result of expanding the square: x^2 + 4x + 4 = (x+2)^2

We'll re-write the integral:

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

We'll use the techinque of changing the variable.

For this reason we'll substitute x+2 by t.

x+2 = t

We'll differentiate both sides:

(x+2)'dx = dt

So, dx = dt

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

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

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

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

But t = x+2

Int dx/(x+2)^2 = -1/(x+2) + C** **

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