3.5 Q-31

Question Statement

Evaluate the integral:

$$\int \frac{3 x^{3}+4 x^{2}+9 x+5}{\left(x^{2}+x+1\right)\left(x^{2}+2x+3\right)} , dx$$

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Background and Explanation

To solve this integral, we need to break the rational function into simpler fractions using partial fraction decomposition. This technique involves expressing the given rational function as a sum of simpler fractions that are easier to integrate.

We also use integration by substitution for certain terms once the decomposition is done.

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Solution

We start by setting up the partial fraction decomposition:

$$\frac{3x^3 + 4x^2 + 9x + 5}{(x^2 + x + 1)(x^2 + 2x + 3)} = \frac{Ax + B}{x^2 + x + 1} + \frac{Cx + D}{x^2 + 2x + 3}$$

Step 1: Multiply both sides by the denominator $(x^2 + x + 1)(x^2 + 2x + 3)$

This gives us:

$$3x^3 + 4x^2 + 9x + 5 = (Ax + B)(x^2 + 2x + 3) + (Cx + D)(x^2 + x + 1)$$

Expanding both sides:

• The first part:

$$(Ax + B)(x^2 + 2x + 3) = A(x^3 + 2x^2 + 3x) + B(x^2 + 2x + 3)$$

• The second part:

$$(Cx + D)(x^2 + x + 1) = C(x^3 + x^2 + x) + D(x^2 + x + 1)$$

Thus, combining both sides:

$$3x^3 + 4x^2 + 9x + 5 = (A + C)x^3 + (2A + B + C + D)x^2 + (3A + 2B + C + D)x + (3B + D)$$

Step 2: Equate coefficients of like powers of $x$

From comparing the powers of $x$ on both sides, we get the following system of equations:

• For $x^3$:
  $A + C = 3 \quad \text{(Equation a)}$

• For $x^2$:
  $2A + B + C + D = 4 \quad \text{(Equation b)}$

• For $x$:
  $3A + 2B + C + D = 9 \quad \text{(Equation c)}$

• For constant terms:
  $3B + D = 5 \quad \text{(Equation d)}$

Step 3: Solve the system of equations

Substituting $C = 3 - A$ into the equations:

• From Equation b: $2A + B + (3 - A) + D = 4 \quad \Rightarrow \quad A + B + D = 1$

• From Equation c: $3A + 2B + (3 - A) + D = 9 \quad \Rightarrow \quad 2A + 2B + D = 6$

Multiply Equation e by 2 and subtract from Equation f:

$$2A + 2B + D = 6 2A + 2B + 2D = 2$$

Subtracting:

$$-D = 4 \quad \Rightarrow \quad D = -4$$

Now substitute $D = -4$ into Equation d:

$$3B + (-4) = 5 \quad \Rightarrow \quad 3B = 9 \quad \Rightarrow \quad B = 3$$

Substituting $B = 3$ and $D = -4$ into Equation e:

$$A + 3 - 4 = 1 \quad \Rightarrow \quad A = 2$$

Finally, substituting $A = 2$ into Equation a:

$$A + C = 3 \quad \Rightarrow \quad 2 + C = 3 \quad \Rightarrow \quad C = 1$$

Step 4: Rewriting the integral with the values of $A$, $B$, $C$, and $D$

We can now express the original integral as:

$$\int \frac{3x^3 + 4x^2 + 9x + 5}{(x^2 + x + 1)(x^2 + 2x + 3)} , dx = \int \frac{2x + 3}{x^2 + x + 1} , dx + \int \frac{x - 4}{x^2 + 2x + 3} , dx$$

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Key Formulas or Methods Used

• Partial Fraction Decomposition: Breaking down a rational function into simpler fractions for easier integration.
• Substitution: Used to handle integrals of the form $\int \frac{dx}{ax^2 + bx + c}$ by completing the square.

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Summary of Steps

1. Set up the partial fraction decomposition.
2. Multiply both sides by the denominator and expand.
3. Equate the coefficients of like powers of $x$.
4. Solve the resulting system of equations for the constants $A$, $B$, $C$, and $D$.
5. Substitute the values of $A$, $B$, $C$, and $D$ back into the original integral.
6. Integrate the resulting simpler fractions.