The task is to compute the integral:

$\int \frac{2x - 3}{x^2 - x - 3} \, dx$

The first instruction suggests substituting a variable for simplification. Let me guide you step by step.

Step 1: Factorize the denominator

The denominator is $x^2 - x - 3$. Let's factorize it:

$x^2 - x - 3 = (x - 3)(x + 1)$

Now, rewrite the integral:

$\int \frac{2x - 3}{(x - 3)(x + 1)} \, dx$

Step 2: Use partial fractions

Decompose the fraction into partial fractions:

$\frac{2x - 3}{(x - 3)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x + 1}$

To find $A$ and $B$, solve:

$2x - 3 = A(x + 1) + B(x - 3)$

Expand:

$2x - 3 = Ax + A + Bx - 3B$

Group terms:

$2x - 3 = (A + B)x + (A - 3B)$

Equate coefficients:

1. $A + B = 2$
2. $A - 3B = -3$

Solve this system of equations:

1. From $A + B = 2$, we get $A = 2 - B$.
2. Substitute $A = 2 - B$ into $A - 3B = -3$: $(2 - B) - 3B = -3 \implies 2 - 4B = -3 \implies -4B = -5 \implies B = \frac{5}{4}$
3. Substitute $B = \frac{5}{4}$ into $A = 2 - B$: $A = 2 - \frac{5}{4} = \frac{8}{4} - \frac{5}{4} = \frac{3}{4}$

Thus, the decomposition is:

$\frac{2x - 3}{(x - 3)(x + 1)} = \frac{\frac{3}{4}}{x - 3} + \frac{\frac{5}{4}}{x + 1}$

Step 3: Rewrite the integral

Split the integral:

$\int \frac{2x - 3}{(x - 3)(x + 1)} \, dx = \frac{3}{4} \int \frac{1}{x - 3} \, dx + \frac{5}{4} \int \frac{1}{x + 1} \, dx$

Compute the integrals:

1. $\int \frac{1}{x - 3} \, dx = \ln|x - 3|$
2. $\int \frac{1}{x + 1} \, dx = \ln|x + 1|$

Thus, the result is:

$\frac{3}{4} \ln|x - 3| + \frac{5}{4} \ln|x + 1| + C$

Final Answer:

$\int \frac{2x - 3}{x^2 - x - 3} \, dx = \frac{3}{4} \ln|x - 3| + \frac{5}{4} \ln|x + 1| + C$

Would you like me to clarify or expand on any step?

Related Questions:

1. How do you factorize quadratic expressions like $x^2 - x - 3$?
2. What are partial fractions, and how are they useful in integration?
3. Why is the substitution step crucial in solving integrals of rational functions?
4. Can you generalize this method to solve similar integrals?
5. How do logarithmic properties simplify the integration process?

Tip:

When solving rational integrals, always check if partial fraction decomposition applies—it often simplifies the problem significantly!