To evaluate the indefinite integral

$\int \frac{x + 5}{x^2 + 10x + 6} \, dx,$

we can use partial fraction decomposition. Let's go through the solution step-by-step:

1. Factor the denominator: The denominator can be factored as follows: $x^2 + 10x + 6 = (x + 1)(x + 6).$

2. Set up the partial fraction decomposition: Since the integrand has a proper rational function, we can express it as: $\frac{x + 5}{(x + 1)(x + 6)} = \frac{A}{x + 1} + \frac{B}{x + 6},$ where $A$ and $B$ are constants to be determined.

3. Solve for $A$ and $B$: Multiply both sides by $(x + 1)(x + 6)$ to clear the denominator: $x + 5 = A(x + 6) + B(x + 1).$ Expanding the right side gives: $x + 5 = Ax + 6A + Bx + B.$ Combine like terms: $x + 5 = (A + B)x + (6A + B).$ By comparing coefficients, we get the following system of equations: $A + B = 1,$ $6A + B = 5.$ Solve for $A$ and $B$:

   • From the first equation, $B = 1 - A$.
   • Substitute $B = 1 - A$ into the second equation: $6A + (1 - A) = 5.$ Simplify: $5A + 1 = 5,$ $5A = 4,$ $A = \frac{4}{5}.$
   • Substitute $A = \frac{4}{5}$ back into $B = 1 - A$: $B = 1 - \frac{4}{5} = \frac{1}{5}.$

   So, $A = \frac{4}{5}$ and $B = \frac{1}{5}$.

4. Rewrite the integrand: Substitute $A$ and $B$ into the partial fraction decomposition: $\frac{x + 5}{(x + 1)(x + 6)} = \frac{\frac{4}{5}}{x + 1} + \frac{\frac{1}{5}}{x + 6}.$

   This simplifies to: $\frac{x + 5}{(x + 1)(x + 6)} = \frac{4}{5} \cdot \frac{1}{x + 1} + \frac{1}{5} \cdot \frac{1}{x + 6}.$

5. Integrate each term: $\int \frac{x + 5}{x^2 + 10x + 6} \, dx = \int \frac{4}{5} \cdot \frac{1}{x + 1} \, dx + \int \frac{1}{5} \cdot \frac{1}{x + 6} \, dx.$ This becomes: $= \frac{4}{5} \ln |x + 1| + \frac{1}{5} \ln |x + 6| + C.$

6. Combine terms (optional): $= \ln |(x + 1)^{4/5} (x + 6)^{1/5}| + C.$

Thus, the final answer is: $\int \frac{x + 5}{x^2 + 10x + 6} \, dx = \frac{4}{5} \ln |x + 1| + \frac{1}{5} \ln |x + 6| + C.$

Would you like more details on any step, or do you have further questions?

Here are five related questions to expand your understanding:

1. How would you handle this integral if the numerator had a higher power than the denominator?
2. What would the solution look like if the denominator couldn't be factored easily?
3. Can you explain why partial fraction decomposition is useful for integration?
4. How would this problem change if it were a definite integral with given bounds?
5. Can you apply partial fraction decomposition to solve integrals with complex roots?

Tip: For integration problems, always check if the integrand can be simplified or factored.