The integral you have provided is:

$\int \frac{x^2 + 4x - 43}{(x + 4)(x + 11)(9x - 4)} \, dx$

To solve this, we will use partial fraction decomposition, expressing the rational function as a sum of simpler fractions. Let’s break this down into the following form:

$\frac{x^2 + 4x - 43}{(x + 4)(x + 11)(9x - 4)} = \frac{A}{x + 4} + \frac{B}{x + 11} + \frac{C}{9x - 4}$

Step-by-Step Process:

1. Set up the equation:

$x^2 + 4x - 43 = A(9x - 4)(x + 11) + B(9x - 4)(x + 4) + C(x + 4)(x + 11)$

Expand both sides to match coefficients of like powers of $x$.

2. Solve for A, B, and C: Multiply out both sides, equate the coefficients of $x^2$, $x$, and constant terms, and solve the system of equations.

3. Integrate each term: Once the values of $A$, $B$, and $C$ are determined, the integral becomes the sum of simpler integrals: $\int \frac{A}{x + 4} dx + \int \frac{B}{x + 11} dx + \int \frac{C}{9x - 4} dx$

Each of these is a standard logarithmic integral of the form $\int \frac{1}{x + a} \, dx = \ln|x + a|$.

I will now calculate the values of $A$, $B$, and $C$ and provide the complete solution. Let me compute this for you.The partial fraction decomposition yields the following values for $A$, $B$, and $C$:

$A = \frac{43}{280}, \quad B = \frac{34}{721}, \quad C = -\frac{3323}{4120}$

Now, we can express the original integrand as:

$\frac{43}{280} \cdot \frac{1}{x + 4} + \frac{34}{721} \cdot \frac{1}{x + 11} - \frac{3323}{4120} \cdot \frac{1}{9x - 4}$

Integrating each term:

1. $\int \frac{43}{280} \cdot \frac{1}{x + 4} \, dx = \frac{43}{280} \ln|x + 4|$
2. $\int \frac{34}{721} \cdot \frac{1}{x + 11} \, dx = \frac{34}{721} \ln|x + 11|$
3. For the third term, we need to account for the $9x - 4$ in the denominator, which involves a substitution $u = 9x - 4$: $\int -\frac{3323}{4120} \cdot \frac{1}{9x - 4} \, dx = -\frac{3323}{4120 \cdot 9} \ln|9x - 4| = -\frac{3323}{37080} \ln|9x - 4|$

Final result:

$\int \frac{x^2 + 4x - 43}{(x + 4)(x + 11)(9x - 4)} \, dx = \frac{43}{280} \ln|x + 4| + \frac{34}{721} \ln|x + 11| - \frac{3323}{37080} \ln|9x - 4| + C$

Would you like more details or have any questions about this result?

Related Questions:

1. What is partial fraction decomposition, and why is it used in integration?
2. How do you expand and simplify polynomial expressions?
3. What are logarithmic integrals and their basic properties?
4. How can we integrate rational functions with repeated linear factors?
5. What is the importance of substitution in integration?

Tip:

When handling rational functions, always check if the degree of the numerator is lower than the denominator before applying partial fractions.