Questions: Part 1. Use the method of partial fraction decomposition to re-write the integrand as the sum of simpler rational functions that can be easily antidifferentiated. [ int frac100 x^2+925 x+399(8 x+5)(x+11)(2 x+8) d x =int ] Part 2. Evaluate the given indefinite integral by evaluating the integral you found in Part 1. above. [ int frac100 x^2+925 x+399(8 x+5)(x+11)(2 x+8) d x = ]

Solution Steps

1. Partial Fraction Decomposition: Decompose the given rational function into simpler fractions. This involves expressing the integrand as a sum of fractions with unknown coefficients.
2. Solve for Coefficients: Determine the values of the unknown coefficients by equating the original numerator to the expanded form of the decomposed fractions.
3. Integrate Each Term: Integrate each of the simpler fractions obtained from the decomposition.

We start with the integrand

\[ \frac{100 x^{2} + 925 x + 399}{(8 x + 5)(x + 11)(2 x + 8)} \]

Using partial fraction decomposition, we can express this as:

\[ \frac{100 x^{2} + 925 x + 399}{(8 x + 5)(x + 11)(2 x + 8)} = -\frac{2}{8 x + 5} + \frac{2}{x + 11} + \frac{9}{2(x + 4)} \]

Next, we integrate each term separately:

1. For the first term:

\[ \int -\frac{2}{8 x + 5} \, dx = -\frac{1}{4} \log(8 x + 5) \]

2. For the second term:

\[ \int \frac{2}{x + 11} \, dx = 2 \log(x + 11) \]

3. For the third term:

\[ \int \frac{9}{2(x + 4)} \, dx = \frac{9}{2} \log(x + 4) \]

Combining these results, we have:

\[ \int \frac{100 x^{2} + 925 x + 399}{(8 x + 5)(x + 11)(2 x + 8)} \, dx = -\frac{1}{4} \log(8 x + 5) + 2 \log(x + 11) + \frac{9}{2} \log(x + 4) + C \]

Final Answer