# Functions.Determine the antiderivative of the function f(x)=(x+1)/(x^2+2x)

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The integral of f(x)=(x+1)/(x^2+2x) can be found using substitution.

If y = x^2 + 2x

dy/dx = 2x + 2 = 2(x + 1)

=> (1/2)dy = (x + 1)dx

Int[(x+1)/(x^2+2x) dx]

=> Int[(1/2)*(1/y) dy]

=> (1/2)ln|y| +C

substitute y = x^2 + 2x

=> (1/2)*ln|x^2 + 2x| + C

**The required integral of f(x)=(x+1)/(x^2+2x) is (1/2)*ln|x^2 + 2x| + C**

We know that the antiderivative of a function is the indefinite integral of the given function:

Int f(x)dx = Int (x+1)dx/(x^2+2x)

We notice that if we'll differentiate the denominator of the function, we'll get the numerator multiplied by 2.

We'll substitute the denominator by t.

x^2+2x = t

We'll differentiate both sides:

(2x + 2)dx = dt

We'll divide by 2:

(x + 1)dx = dt/2

We'll re-write the integral in t:

Int f(x)dx = Int dt/2t = (1/2)*ln |t| + C

Int f(x)dx = (1/2)*ln |x^2+2x| + C

**The antiderivative of f(x) is: ln sqrt (x^2+2x) + C**