Find the antiderivative of the function f(x)=(x+1)/(x^2+2x) .
We have to find the anti derivative of f(x)=(x+1)/(x^2+2x)
let x^2 + 2x = t
=> dt/dx = 2x + 2 = 2(x + 1)
Int [(x+1)/(x^2+2x) dx]
=> Int [ (1/2t) dt]
=> (1/2) ln t + C
replace t with x^2 + 2x
=> (1/2) ln (x^2 + 2x) + C
The required result is (1/2) ln (x^2 + 2x) + C
We'll determine 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
Int f(x)dx = ln sqrt (x^2+2x) + C