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The given integral is:

\[
\int \ln(x^2 + 16x + 60) \, dx.
\]

To solve this, we can use integration by parts. The integral is of the form:

\[
\int u \, dv = u v - \int v \, du.
\]

We'll let \( u = \ln(x^2 + 16x + 60) \) and find the differential \( du \). Next, \( dv = dx \), which is straightforward to integrate as \( v = x \).

Let's compute this step-by-step:

1. Set \( u = \ln(x^2 + 16x + 60) \). Then the derivative \( du \) is found using the chain rule:
   \[
   du = \frac{d}{dx} \ln(x^2 + 16x + 60) \cdot dx = \frac{2x + 16}{x^2 + 16x + 60} \, dx.
   \]

2. Now, \( dv = dx \), so integrating this gives \( v = x \).

Using integration by parts, we now have:

\[
\int \ln(x^2 + 16x + 60) \, dx = x \ln(x^2 + 16x + 60) - \int x \cdot \frac{2x + 16}{x^2 + 16x + 60} \, dx.
\]

This remaining integral can be simplified. Let's solve it:

The integral we are left with is:

\[
\int \frac{x(2x + 16)}{x^2 + 16x + 60} \, dx.
\]

Notice that the numerator can be rewritten as \( 2(x^2 + 16x) \), which allows us to break the fraction:

\[
\int \frac{2(x^2 + 16x) + 120}{x^2 + 16x + 60} \, dx = 2 \int \frac{x^2 + 16x + 60}{x^2 + 16x + 60} \, dx - 120 \int \frac{dx}{x^2 + 16x + 60}.
\]

The first integral simplifies to:

\[
2 \int dx = 2x.
\]

For the second integral \( \int \frac{dx}{x^2 + 16x + 60} \), we can complete the square for the quadratic expression \( x^2 + 16x + 60 \):

\[
x^2 + 16x + 60 = (x + 8)^2 - 4.
\]

This becomes a standard form for the integral:

\[
\int \frac{dx}{(x + 8)^2 - 4} = \frac{1}{2} \ln \left| \frac{x + 8 - 2}{x + 8 + 2} \right| = \frac{1}{2} \ln \left| \frac{x + 6}{x + 10} \right|.
\]

Thus, the solution to the original integral is:

\[
x \ln(x^2 + 16x + 60) - 2x - 60 \cdot \frac{1}{2} \ln \left| \frac{x + 6}{x + 10} \right| + C,
\]

where \( C \) is the constant of integration.

Would you like any further clarifications or details?

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Here are 5 related questions to deepen your understanding:

1. How does integration by parts apply to logarithmic integrals?
2. Can you explain the process of completing the square in more detail?
3. Why do we break the fraction during the integration process?
4. How does the form of the denominator affect the type of integral?
5. What are other applications of completing the square in integration?

**Tip:** Always look for symmetry or simplifications like completing the square to handle quadratic expressions in integrals efficiently.

Evaluate the integral of ln(x^2 + 16x + 60) with respect to x.

This solution explains how to evaluate the integral of ln(x^2 + 16x + 60) using integration by parts. It covers key calculus techniques including logarithmic differentiation, completing the square, and the use of standard integrals.

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