This exercise involves calculating the exact value of a given definite integral using partial fraction decomposition. Let's break down the task step by step:

Given integral:

$\int_0^1 \frac{x}{(x + 8)(x + 7)^2} \, dx$

The goal is to use partial fraction decomposition of the integrand.

Partial fraction decomposition:

The form of the decomposition is given by: $\frac{x}{(x + 8)(x + 7)^2} = \frac{A}{x + 8} + \frac{B}{x + 7} + \frac{C}{(x + 7)^2}$

Step a) Finding coefficients $A$, $B$, and $C$:

To find $A$, $B$, and $C$, multiply both sides by $(x + 8)(x + 7)^2$ to clear the denominators: $x = A(x + 7)^2 + B(x + 8)(x + 7) + C(x + 8)$

Now expand both sides and match coefficients for powers of $x$.

1. Expanding $A(x + 7)^2 = A(x^2 + 14x + 49)$
2. Expanding $B(x + 8)(x + 7) = B(x^2 + 15x + 56)$
3. Expanding $C(x + 8) = Cx + 8C$

The left-hand side is just $x$, which means the right-hand side must simplify to $x$.

Thus, we have: $x = A(x^2 + 14x + 49) + B(x^2 + 15x + 56) + Cx + 8C$

Now collect the powers of $x$:

• $x^2$ terms: $A + B = 0$
• $x^1$ terms: $14A + 15B + C = 1$
• Constant terms: $49A + 56B + 8C = 0$

Solve this system of equations for $A$, $B$, and $C$.

1. From $A + B = 0$, we have $B = -A$.
2. Substitute into the second equation: $14A + 15(-A) + C = 1 \Rightarrow -A + C = 1 \Rightarrow C = A + 1$.
3. Substitute into the third equation: $49A + 56(-A) + 8(A + 1) = 0 \Rightarrow -7A + 8A + 8 = 0 \Rightarrow A = -8$.
4. Hence, $B = 8$ and $C = -7$.

Thus, the coefficients are: $A = -8, \quad B = 8, \quad C = -7$

Step b) The integral function with respect to $x$:

Using the partial fraction decomposition, the integrand can now be written as: $\frac{-8}{x + 8} + \frac{8}{x + 7} - \frac{7}{(x + 7)^2}$

Now, integrate each term: $\int \left( \frac{-8}{x + 8} + \frac{8}{x + 7} - \frac{7}{(x + 7)^2} \right) \, dx$

1. The first term gives: $-8 \ln|x + 8|$.
2. The second term gives: $8 \ln|x + 7|$.
3. The third term gives: $\frac{7}{x + 7}$.

Thus, the integral function (with the constant of integration omitted) is: $-8 \ln|x + 8| + 8 \ln|x + 7| - \frac{7}{x + 7}$

Step c) The exact value of the definite integral:

To find the exact value of the definite integral, evaluate this result at the limits $x = 1$ and $x = 0$.

For $x = 1$: $-8 \ln(1 + 8) + 8 \ln(1 + 7) - \frac{7}{1 + 7} = -8 \ln 9 + 8 \ln 8 - \frac{7}{8}$

For $x = 0$: $-8 \ln(0 + 8) + 8 \ln(0 + 7) - \frac{7}{0 + 7} = -8 \ln 8 + 8 \ln 7 - 1$

Now subtract the result at $x = 0$ from the result at $x = 1$: $\left( -8 \ln 9 + 8 \ln 8 - \frac{7}{8} \right) - \left( -8 \ln 8 + 8 \ln 7 - 1 \right)$

Simplifying: $-8 \ln 9 + 8 \ln 8 - \frac{7}{8} + 8 \ln 8 - 8 \ln 7 + 1$ $= 8(\ln 8 - \ln 9 - \ln 7) + 1 - \frac{7}{8}$ $= 8 \ln \left( \frac{8^2}{9 \cdot 7} \right) + 1 - \frac{7}{8}$

This is the exact value of the definite integral.

Do you want more details on any step, or have any questions?

Here are 5 related questions to consider:

1. How would the result change if the limits of integration were different?
2. What is the importance of partial fraction decomposition in integral calculus?
3. Can this method be applied to more complex rational functions?
4. How does the choice of logarithmic terms affect integration results?
5. How would the integral change if $x^2$ were in the numerator?

Tip: Always check if the function allows partial fraction decomposition before applying it!