Question

Use partial fractions to find the indefinite integral. $$ \int \frac{x+1}{x^{2}+4 x+3} d x $$

Short answer

The solution to the integral is \(\ln|x+3| + C\).

Step-by-step solution

Factoring the denominator

First, factorize the denominator into two linear factors: \(x^{2}+4x+3\) can be factored into \((x+1)(x+3)\)

Writing the fraction as a sum of partial fractions

Rewrite \(\frac{x+1}{x^{2}+4 x+3}\) as \(\frac{A}{x+1}+\frac{B}{x+3}\) for some constants A and B. Multiplying through by the common denominator gives \(x+1=A(x+3)+B(x+1)\)

Solving for A and B

To determine A and B, you can set up a system of equations by equating the coefficients on each side of the equation. This yields the system: A+B=1, 3A+B=1. Solving the system gives A=0, B=1.

Integrating

Replace A and B in the partial fraction decomposition and integrate term by term to obtain: \(\int \frac{0}{x+1} dx + \int \frac{1}{x+3} dx = 0 + \ln|x+3| + C\) where C is the constant of integration.