To solve the integral:

$I = \int \frac{3x^2 + x + 23}{(x - 4)(x^2 + 9)} \, dx$

Step 1: Perform partial fraction decomposition

We seek to express the integrand in the form:

$\frac{3x^2 + x + 23}{(x - 4)(x^2 + 9)} = \frac{A}{x - 4} + \frac{Bx + C}{x^2 + 9}$

Multiplying both sides by $(x - 4)(x^2 + 9)$ gives:

$3x^2 + x + 23 = A(x^2 + 9) + (Bx + C)(x - 4)$

Step 2: Expand and collect terms

Expand $A(x^2 + 9)$

$A(x^2 + 9) = Ax^2 + 9A$

Expand $(Bx + C)(x - 4)$

$(Bx + C)(x - 4) = Bx^2 - 4Bx + Cx - 4C$

Now combine all terms:

$3x^2 + x + 23 = Ax^2 + 9A + Bx^2 - 4Bx + Cx - 4C$

Group terms by powers of $x$:

$( A + B )x^2 + ( -4B + C )x + ( 9A - 4C ) = 3x^2 + x + 23$

Step 3: Solve for coefficients

Coefficient of $x^2$

A + B = 3 \tag{1}

Coefficient of $x$

-4B + C = 1 \tag{2}

Constant term

9A - 4C = 23 \tag{3}

Solve this system of equations:

From Equation (1)

$B = 3 - A$

Substitute into Equation (2)

-4(3 - A) + C = 1 \implies -12 + 4A + C = 1 \implies C = 13 - 4A \tag{4}

Substitute $B = 3 - A$ and $C = 13 - 4A$ into Equation (3)

$9A - 4(13 - 4A) = 23$

Simplify:

$9A - 52 + 16A = 23 \implies 25A = 75 \implies A = 3$

Find $B$ and $C$

$B = 3 - A = 0 \quad \text{and} \quad C = 13 - 4(3) = 1$

Thus, the decomposition becomes:

$\frac{3x^2 + x + 23}{(x - 4)(x^2 + 9)} = \frac{3}{x - 4} + \frac{x + 1}{x^2 + 9}$

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Step 4: Integrate each term

Integral of $\frac{3}{x - 4}$

$\int \frac{3}{x - 4} \, dx = 3 \ln|x - 4|$

Integral of $\frac{x}{x^2 + 9}$

Let $u = x^2 + 9$, so $du = 2x dx$. Therefore,

$\int \frac{x}{x^2 + 9} dx = \frac{1}{2} \ln|x^2 + 9|$

Integral of $\frac{1}{x^2 + 9}$

This is a standard integral:

$\int \frac{1}{x^2 + 9} dx = \frac{1}{3} \arctan\left(\frac{x}{3}\right)$

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Step 5: Combine results

$I = 3 \ln|x - 4| + \frac{1}{2} \ln|x^2 + 9| + \frac{1}{3} \arctan\left(\frac{x}{3}\right) + C$

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Would you like more details or clarifications on any step? 😊

Related Questions

1. How does partial fraction decomposition work for repeated roots?
2. Can we use substitution instead of partial fractions here?
3. What are the geometric interpretations of logarithmic and arctan integrals?
4. How does the arctan function relate to trigonometry in integrals?
5. What are common mistakes in solving rational function integrals?

Tip:

Always double-check coefficient matching carefully when solving partial fraction decompositions!