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

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

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