3.5 Q-17

Question Statement

Evaluate the integral:

$$\int \frac{x^{3}+22x^{2}+14x-17}{(x-3)(x+2)^{3}} , dx$$

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Background and Explanation

To solve this integral, we will use partial fraction decomposition. The integrand is a rational function, and the denominator is factored into linear and quadratic terms. The general approach is to express the rational function as a sum of simpler fractions, which can then be integrated individually. The terms we will use are based on the factors in the denominator: $(x-3)$ and $(x+2)^3$.

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Solution

Step 1: Set up the Partial Fraction Decomposition

We assume that the integrand can be written as:

$$\frac{x^{3}+22x^{2}+14x-17}{(x-3)(x+2)^{3}} = \frac{A}{x-3} + \frac{B}{x+2} + \frac{C}{(x+2)^2} + \frac{D}{(x+2)^3}$$

Multiply both sides by $(x-3)(x+2)^3$ to clear the denominators:

$$x^{3} + 22x^{2} + 14x - 17 = A(x+2)^3 + B(x-3)(x+2)^2 + C(x-3)(x+2) + D(x-3)$$

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Step 2: Solve for Constants $A$, $B$, $C$, and $D$

Finding $A$:

Substitute $x = 3$ into the equation to eliminate the terms involving $B$, $C$, and $D$:

$$(3)^3 + 22(3)^2 + 14(3) - 17 = A(3+2)^3$$

This simplifies to:

$$27 + 198 + 42 - 17 = A(5)^3$$ $$250.4 = 125A \quad \Rightarrow \quad A = 2$$

Finding $D$:

Substitute $x = -2$ into the equation to eliminate the terms involving $A$, $B$, and $C$:

$$(-2)^3 + 22(-2)^2 + 14(-2) - 17 = D(-2-3)$$

This simplifies to:

$$-8 + 88 - 28 - 17 = -5D$$ $$35 = -5D \quad \Rightarrow \quad D = -7$$

Finding $B$ and $C$:

Now, we compare the coefficients of like powers of $x$. From the $x^3$ term:

$$1 = A + B \quad \Rightarrow \quad B = 1 - A = 1 - 2 = -1$$

Next, use the constant term to find $C$:

$$17 = 8A - 1B - 6C - 3D$$

Substitute the known values for $A$, $B$, and $D$:

$$17 = 8(2) - 1(-1) - 6C - 3(-7)$$

This simplifies to:

$$17 = 16 + 1 - 6C + 21 \quad \Rightarrow \quad 17 = 38 - 6C$$

Solve for $C$:

$$-6C = 17 - 38 \quad \Rightarrow \quad -6C = -21 \quad \Rightarrow \quad C = 11$$

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Step 3: Substitute Back into the Decomposition

Now that we have the values for $A$, $B$, $C$, and $D$, we can rewrite the integrand as:

$$\frac{x^{3}+22x^{2}+14x-17}{(x-3)(x+2)^3} = \frac{2}{x-3} - \frac{1}{x+2} + \frac{11}{(x+2)^2} - \frac{7}{(x+2)^3}$$

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Step 4: Integrate Each Term

Now, integrate each term separately:

$$\int \frac{2}{x-3} , dx = 2 \ln |x-3|$$ $$\int \frac{-1}{x+2} , dx = -\ln |x+2|$$ $$\int \frac{11}{(x+2)^2} , dx = \frac{-11}{x+2}$$ $$\int \frac{-7}{(x+2)^3} , dx = \frac{7}{2(x+2)^2}$$

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Key Formulas or Methods Used

• Partial Fraction Decomposition: We split the rational function into simpler fractions that are easier to integrate.
• Integration of Rational Functions: The standard formulas for the integrals of terms like $\frac{1}{x-a}$ and $\frac{1}{(x-a)^n}$.

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Summary of Steps

1. Set up the partial fraction decomposition based on the factors of the denominator.
2. Multiply through by the denominator to eliminate fractions.
3. Substitute values of $x$ to solve for the constants $A$, $B$, $C$, and $D$.
4. Rewrite the rational function using the constants found.
5. Integrate each term separately and combine the results.

The final answer is:

$$2 \ln |x-3| - \ln |x+2| - \frac{11}{x+2} + \frac{7}{2(x+2)^2} + C$$