Science:Math Exam Resources/Courses/MATH101 C/April 2025/Question 02 (b)/Solution 1

We first recognize that the denominator can be factored as follows $x^{2} + 3 x + 2 = (x + 1) (x + 2) .$ Now, in order to simplify this expression, we can use partial fraction decomposition, i.e., we want to find $A, B$ so that $\frac{5 x + 8}{(x + 1) (x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2} .$ Since the right-hand side is equal to $\frac{A (x + 2) + B (x + 1)}{(x + 1) (x + 2)}$ , we must have $5 x + 8 = A (x + 2) + B (x + 1) = (A + B) x + 2 A + B$ , as an equation that holds for all $x$ . Therefore, we must have ${\begin{cases} A + B & = 5 \\ 2 A + B & = 8 \end{cases}$ From the first equation, $B = 5 - A$ . Substituting this into the second equation yields $A = 3$ , so we have $3 + B = 5, B = 2$ . We can therefore rewrite the integrand and find the antiderivative as follows:

$\int \frac{5 x + 8}{(x + 1) (x + 2)} d x = \int (\frac{3}{x + 1} + \frac{2}{x + 2}) d x = 3 \log | x + 1 | + 2 \log | x + 2 | + C .$