Use partial fraction dec $$ \int \frac{6 v-55}{9 v^{2}-9 v-40} d v= $$

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To evaluate the integral

$$
\int \frac{6v - 55}{9v^2 - 9v - 40} \, dv,
$$

we'll use partial fraction decomposition.

### Step 1: Factor the Denominator

First, factor the quadratic denominator:

$$
9v^2 - 9v - 40.
$$

We look for two numbers that multiply to $$9 \times (-40) = -360$$ and add up to $$-9$$. The numbers $$-24$$ and $$15$$ satisfy this:

$$
9v^2 - 24v + 15v - 40 = 3v(3v - 8) + 5(3v - 8) = (3v + 5)(3v - 8).
$$

So,

$$
9v^2 - 9v - 40 = (3v + 5)(3v - 8).
$$

### Step 2: Partial Fraction Decomposition

Express the integrand as:

$$
\frac{6v - 55}{(3v + 5)(3v - 8)} = \frac{A}{3v + 5} + \frac{B}{3v - 8}.
$$

Multiplying both sides by $$(3v + 5)(3v - 8)$$ gives:

$$
6v - 55 = A(3v - 8) + B(3v + 5).
$$

To find $$A$$ and $$B$$, choose strategic values for $$v$$:

1. Let $$v = -\frac{5}{3}$$:
   $$
   6\left(-\frac{5}{3}\right) - 55 = A\left(3\left(-\frac{5}{3}\right) - 8\right) \Rightarrow -10 - 55 = A(-13) \Rightarrow A = 5.
   $$

2. Let $$v = \frac{8}{3}$$:
   $$
   6\left(\frac{8}{3}\right) - 55 = B\left(3\left(\frac{8}{3}\right) + 5\right) \Rightarrow 16 - 55 = B(13) \Rightarrow B = -3.
   $$

Thus, the decomposition is:

$$
\frac{6v - 55}{(3v + 5)(3v - 8)} = \frac{5}{3v + 5} - \frac{3}{3v - 8}.
$$

### Step 3: Integrate

Now, integrate term by term:

$$
\int \left( \frac{5}{3v + 5} - \frac{3}{3v - 8} \right) dv = \frac{5}{3} \ln|3v + 5| - \ln|3v - 8| + C,
$$

where $$C$$ is the constant of integration.

**Final Answer:**

After simplifying, an antiderivative is (5⁄3) ln|3v + 5| minus ln|3v – 8| plus C. Thus,

∫ = (5/3) ln|3v+5| − ln|3v−8| + C
The integral simplifies to: $$ \frac{5}{3} \ln|3v + 5| - \ln|3v - 8| + C $$

Use partial fraction dec \( \int \frac{6 v-55}{9 v^{2}-9 v-40} d v= \)

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