3.5 Q-3

Question Statement

Evaluate the integral:
$\int \frac{x^{2}+3x-34}{x^{2}+2x-15} , dx$

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Background and Explanation

This is a rational function integral where the degree of the numerator is equal to the degree of the denominator. To simplify, we divide the numerator by the denominator and then decompose the resulting fraction into partial fractions. Familiarity with partial fraction decomposition and logarithmic integration is required.

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Solution

Step 1: Simplify the Rational Function

The denominator is factored as: $x^2 + 2x - 15 = (x + 5)(x - 3)$

We divide the numerator by the denominator: $\frac{x^2 + 3x - 34}{x^2 + 2x - 15} = 1 + \frac{x - 19}{x^2 + 2x - 15}$

This reduces the integral to: $\int \frac{x^2 + 3x - 34}{x^2 + 2x - 15} , dx = \int 1 , dx + \int \frac{x - 19}{x^2 + 2x - 15} , dx$

Step 2: Partial Fraction Decomposition

For the term $\frac{x - 19}{x^2 + 2x - 15}$, we write: $\frac{x - 19}{(x + 5)(x - 3)} = \frac{A}{x + 5} + \frac{B}{x - 3}$

Multiplying through by $(x + 5)(x - 3)$, we get: $x - 19 = A(x - 3) + B(x + 5)$

Step 3: Solve for Coefficients

Expand and group terms: $x - 19 = A(x) - 3A + B(x) + 5B = (A + B)x + (-3A + 5B)$

Equating coefficients:

1. For $x$: $A + B = 1$
2. For constants: $-3A + 5B = -19$

Solve the system of equations:

• From $A + B = 1$, $B = 1 - A$
• Substitute into $-3A + 5B = -19$: $-3A + 5(1 - A) = -19$ $-3A + 5 - 5A = -19$ $-8A = -24 \quad \Rightarrow \quad A = 3$ $B = 1 - A = 1 - 3 = -2$

Step 4: Rewrite the Integral

Substituting $A = 3$ and $B = -2$: $\frac{x - 19}{(x + 5)(x - 3)} = \frac{3}{x + 5} - \frac{2}{x - 3}$

The integral becomes: $\int 1 , dx + \int \frac{3}{x + 5} , dx - \int \frac{2}{x - 3} , dx$

Step 5: Integrate

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

Combining terms: $\int \frac{x^2 + 3x - 34}{x^2 + 2x - 15} , dx = x + 3\ln|x + 5| - 2\ln|x - 3| + C$

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Key Formulas or Methods Used

1. Partial Fraction Decomposition: $\frac{P(x)}{Q(x)} = \frac{A}{\text{factor1}} + \frac{B}{\text{factor2}}$
2. Logarithmic Integration: $\int \frac{1}{x + c} , dx = \ln|x + c|$

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Summary of Steps

1. Factorize the denominator: $(x + 5)(x - 3)$.
2. Simplify the fraction using polynomial division.
3. Decompose the remainder into partial fractions.
4. Solve for coefficients $A$ and $B$.
5. Rewrite and integrate each term individually.
6. Combine results: $\boxed{x + 3\ln|x + 5| - 2\ln|x - 3| + C}$