To solve the integral [tex]\(\int \frac{4x - 5}{(x + 1)(x - 2)} \, dx\)[/tex], we can employ the method of partial fraction decomposition. Here’s a step-by-step breakdown:
1. Express the integral as a partial fraction:
We start by expressing the integrand [tex]\(\frac{4x - 5}{(x + 1)(x - 2)}\)[/tex] as a sum of simpler fractions. Assume that:
[tex]\[
\frac{4x - 5}{(x + 1)(x - 2)} = \frac{A}{x + 1} + \frac{B}{x - 2}
\][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants that we need to determine.
2. Find the constants [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
To determine [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we combine the fractions on the right-hand side:
[tex]\[
\frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x - 2) + B(x + 1)}{(x + 1)(x - 2)}
\][/tex]
Since the denominators are equal, we can equate the numerators:
[tex]\[
4x - 5 = A(x - 2) + B(x + 1)
\][/tex]
Simplify and collect like terms:
[tex]\[
4x - 5 = (A + B)x + (-2A + B)
\][/tex]
Now, we solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex] by comparing the coefficients of [tex]\(x\)[/tex] and the constant terms from both sides of the equation:
[tex]\[
A + B = 4 \quad \text{(coefficient of } x\text{)}
\][/tex]
[tex]\[
-2A + B = -5 \quad \text{(constant term)}
\][/tex]
3. Solve the system of equations:
We solve these simultaneous equations to find [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- From [tex]\(A + B = 4\)[/tex], we can express [tex]\(B\)[/tex] in terms of [tex]\(A\)[/tex]:
[tex]\[
B = 4 - A
\][/tex]
- Substitute [tex]\(B\)[/tex] into [tex]\(-2A + B = -5\)[/tex]:
[tex]\[
-2A + (4 - A) = -5
\][/tex]
[tex]\[
-3A + 4 = -5
\][/tex]
[tex]\[
-3A = -9
\][/tex]
[tex]\[
A = 3
\][/tex]
- Substitute [tex]\(A = 3\)[/tex] back into [tex]\(B = 4 - A\)[/tex]:
[tex]\[
B = 4 - 3 = 1
\][/tex]
4. Rewrite the integrand with [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
Now, substitute the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] back into the partial fraction decomposition:
[tex]\[
\frac{4x - 5}{(x + 1)(x - 2)} = \frac{3}{x + 1} + \frac{1}{x - 2}
\][/tex]
5. Integrate each term separately:
Now, integrate term by term:
[tex]\[
\int \frac{4x - 5}{(x + 1)(x - 2)} \, dx = \int \frac{3}{x + 1} \, dx + \int \frac{1}{x - 2} \, dx
\][/tex]
Integrate each term using the fact that the integral of [tex]\(\frac{1}{x}\)[/tex] is [tex]\(\ln|x|\)[/tex]:
[tex]\[
\int \frac{3}{x + 1} \, dx = 3 \ln|x + 1|
\][/tex]
[tex]\[
\int \frac{1}{x - 2} \, dx = \ln|x - 2|
\][/tex]
6. Combine the results and include the constant of integration:
Therefore, the final result of the integral is:
[tex]\[
\boxed{\ln|x - 2| + 3 \ln|x + 1| + C}
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
