Question

Find the value of \( \int \frac{x+2}{2x^2+6x+5} \,dx \).

Hint:

For rational integrals, factor the denominator and use partial fraction decomposition before integrating.

Answer 1

We start by factoring the denominator: \[ 2x^2 + 6x + 5 = 2(x^2 + 3x + \frac{5}{2}) \] Factoring further: \[ 2x^2 + 6x + 5 = 2(x + 1)(x + \frac{5}{2}). \] Now, use partial fraction decomposition: \[ \frac{x+2}{2(x+1)(x+\frac{5}{2})} = \frac{A}{x+1} + \frac{B}{x+\frac{5}{2}}. \] Multiplying by the denominator: \[ x+2 = A(x+\frac{5}{2}) + B(x+1). \] Solving for \( A \) and \( B \), we integrate each term separately and get: \[ \int \frac{x+2}{2x^2+6x+5} dx = \frac{1}{2} \ln |x+1| - \frac{1}{2} \ln |x+\frac{5}{2}| + C. \]