Evaluate the integral: $\int x \ln (x-8) d x$

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riley6869i riley6869i

02-11-2026
Mathematics

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Evaluate the integral: $\int x \ln (x-8) d x$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T17:54:19-05:00

Set up integration by parts by choosing $u = \ln(x-8)$ and $dv = x \, dx$.
Simplify the resulting rational function $\frac{x^2}{2(x-8)}$ using polynomial division to integrate term by term.
Combine all evaluated components to obtain the result.
The final indefinite integral is $\boxed{(\frac{1}{2}x^2 - 32) \ln(x-8) - \frac{1}{4}x^2 - 4x + C}$.

Explanation

Problem Analysis and Integration Technique Choice

To evaluate the indefinite integral $\int x \ln(x-8) \, dx$, we identify the integrand as a product of two functions, $f(x) = x$ and $g(x) = \ln(x-8)$. The function is defined for $x > 8$. We will use the integration by parts formula, which states that $\int u \, dv = uv - \int v \, du$.

Applying Integration by Parts

We choose $u = \ln(x-8)$ and $dv = x \, dx$.
Differentiating $u$ gives $du = \frac{1}{x-8} \, dx$.
Integrating $dv$ gives $v = \int x \, dx = \frac{1}{2}x^2$.
Applying the integration by parts formula:
$\int x \ln(x-8) \, dx = \left(\frac{1}{2}x^2\right) \ln(x-8) - \int \left(\frac{1}{2}x^2\right) \frac{1}{x-8} \, dx$.

Simplifying and Evaluating the Remaining Integral

We need to simplify the integral $\int \frac{x^2}{2(x-8)} \, dx$. We can rewrite the integrand by polynomial division:
$\frac{x^2}{x-8} = \frac{x^2 - 64 + 64}{x-8} = \frac{(x-8)(x+8) + 64}{x-8} = x + 8 + \frac{64}{x-8}$.
Thus, the integral becomes:
$\int \frac{x^2}{2(x-8)} \, dx = \frac{1}{2} \int \left(x + 8 + \frac{64}{x-8}\right) \, dx$.
Evaluating this term by term gives:
$\frac{1}{2} \left(\frac{1}{2}x^2 + 8x + 64 \ln(x-8)\right) = \frac{1}{4}x^2 + 4x + 32 \ln(x-8)$.

Final Result Compilation

Combining the parts together, we get:
$\int x \ln(x-8) \, dx = \frac{1}{2}x^2 \ln(x-8) - \left(\frac{1}{4}x^2 + 4x + 32 \ln(x-8)\right) + C$.
Grouping the $\ln(x-8)$ terms leads to:
$\left(\frac{1}{2}x^2 - 32\right) \ln(x-8) - \frac{1}{4}x^2 - 4x + C$.
This is the final form of the indefinite integral.

Examples

Integration by parts is essential in fields like physics and engineering, where you often need to calculate work done by variable forces or find centers of mass for objects with varying densities. By breaking down complex integral expressions involving products of functions, you can derive accurate models to optimize performance in mechanical systems or electrical circuits. For instance, determining the energy distribution in a system often requires evaluating integrals of the form found in this problem.