Questions: Evaluate the following integral. [ int fracx^2(x-5)^3 d x ]

Solution Steps

To evaluate the integral \(\int \frac{x^{2}}{(x-5)^{3}} dx\), we can use the method of integration by parts or partial fraction decomposition. In this case, partial fraction decomposition is more suitable. We express the integrand as a sum of simpler fractions and then integrate each term separately.

We start with the integral

\[ \int \frac{x^{2}}{(x-5)^{3}} dx. \]

Using partial fraction decomposition, we express the integrand as:

\[ \frac{x^{2}}{(x-5)^{3}} = \frac{1}{x-5} + \frac{10}{(x-5)^{2}} + \frac{25}{(x-5)^{3}}. \]

Next, we integrate each term separately:

\[ \int \left( \frac{1}{x-5} + \frac{10}{(x-5)^{2}} + \frac{25}{(x-5)^{3}} \right) dx. \]

The integrals yield:

1. \(\int \frac{1}{x-5} dx = \log|x-5|\)
2. \(\int \frac{10}{(x-5)^{2}} dx = -\frac{10}{x-5}\)
3. \(\int \frac{25}{(x-5)^{3}} dx = -\frac{25}{2(x-5)^{2}}\)

Combining these results, we have:

\[ \int \frac{x^{2}}{(x-5)^{3}} dx = \log|x-5| - \frac{10}{x-5} - \frac{25}{2(x-5)^{2}} + C, \]

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

The final result can be expressed as:

\[ \int \frac{x^{2}}{(x-5)^{3}} dx = \frac{75 - 20x}{2(x-5)^{2}} + \log|x-5| + C. \]

Final Answer