To solve the integral

$I = \int \frac{x^3 - 2x^2 + 2x - 7}{x^2 - 2x - 3} \, dx,$

we can proceed with polynomial long division, followed by partial fraction decomposition if necessary.

Step 1: Polynomial Long Division

Divide the numerator $x^3 - 2x^2 + 2x - 7$ by the denominator $x^2 - 2x - 3$.

1. Divide $x^3$ by $x^2$, which gives $x$.
2. Multiply $x$ by the denominator $x^2 - 2x - 3$, resulting in $x^3 - 2x^2 - 3x$.
3. Subtract $(x^3 - 2x^2 - 3x)$ from the original numerator:

$(x^3 - 2x^2 + 2x - 7) - (x^3 - 2x^2 - 3x) = 5x - 7.$

Now, we need to divide the remainder $5x - 7$ by $x^2 - 2x - 3$.

Since the degree of the remainder is lower than that of the denominator, we can now express the integral as:

$I = \int (x + \frac{5x - 7}{x^2 - 2x - 3}) \, dx.$

Step 2: Partial Fraction Decomposition

Factor the denominator $x^2 - 2x - 3$ as $(x - 3)(x + 1)$. So we want to decompose:

$\frac{5x - 7}{(x - 3)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x + 1}.$

Multiply both sides by $(x - 3)(x + 1)$ to get:

$5x - 7 = A(x + 1) + B(x - 3).$

Expanding the right-hand side:

$5x - 7 = A(x + 1) + B(x - 3) = A x + A + B x - 3B = (A + B)x + (A - 3B).$

Now, equate the coefficients of like terms:

1. For $x$-terms: $A + B = 5$,
2. For the constant terms: $A - 3B = -7$.

Solving this system of equations:

• From $A + B = 5$, we have $A = 5 - B$.
• Substituting into $A - 3B = -7$:

$(5 - B) - 3B = -7 \implies 5 - 4B = -7 \implies -4B = -12 \implies B = 3.$

• Substituting $B = 3$ into $A + B = 5$, we get $A = 2$.

Step 3: Rewrite the Integral

Now, we can express the integral as:

$I = \int \left( x + \frac{2}{x - 3} + \frac{3}{x + 1} \right) dx.$

Step 4: Integrate Term by Term

Now integrate each term:

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

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

Thus, the final result is:

$I = \frac{x^2}{2} + 2 \ln |x - 3| + 3 \ln |x + 1| + C,$

where $C$ is the constant of integration.

Final Answer:

$\int \frac{x^3 - 2x^2 + 2x - 7}{x^2 - 2x - 3} \, dx = \frac{x^2}{2} + 2 \ln |x - 3| + 3 \ln |x + 1| + C.$

Would you like more details or clarification on any step?

Here are 5 related questions for further practice:

1. How would you integrate $\frac{x^3 - 4x + 5}{x^2 - 1} \, dx$?
2. What is the general approach to solve an integral involving rational functions?
3. Can you explain how to use synthetic division in similar integration problems?
4. How do you perform partial fraction decomposition for higher-degree polynomials?
5. How would the integral change if the denominator had complex roots?

Tip: Always check if the degree of the numerator is greater than or equal to the denominator for polynomial long division.