\( J = \int \frac { x + 1 } { x ^ { 2 } + 2 x - 5 } d x \)

To solve the integral \( J = \int \frac { x + 1 } { x ^ { 2 } + 2 x - 5 } d x \), we can start by simplifying the integrand. First, we'll factor the denominator. The expression \( x^2 + 2x - 5 \) can be completed into a square: \[ x^2 + 2x - 5 = (x + 1)^2 - 6 \] Now, let's perform polynomial long division if necessary or break down the integral into two manageable parts. Considering \( J = \int \frac{x + 1}{(x + 1)^2 - 6} dx \), it might help to split the fraction via algebraic manipulation: \[ \frac{x + 1}{x^2 + 2x - 5} = \frac{x + 1}{(x + 1)^2 - 6} = \frac{1}{2} \frac{d}{dx} \ln| (x + 1)^2 - 6 | + C \] The integral now simplifies to: \[ J = \frac{1}{2} \ln| (x + 1)^2 - 6 | + C \] So, the integral evaluates elegantly to: \[ J = \frac{1}{2} \ln| x^2 + 2x - 5 | + C \] where \( C \) is the constant of integration.