Questions: ∫ from 4 to 5 (7x-5)/(x^2-x-2) dx

Transcript text: $\int_{4}^{5} \frac{7 x-5}{x^{2}-x-2} d x$

Solution

Solution Steps

To solve the integral $\int_{4}^{5} \frac{7 x-5}{x^{2}-x-2} d x$, we can use the method of partial fraction decomposition to break down the integrand into simpler fractions that are easier to integrate.

Solution Approach

Factor the denominator $x^2 - x - 2$.
Decompose the fraction $\frac{7x - 5}{(x-2)(x+1)}$ into partial fractions.
Integrate each term separately over the interval [4, 5].

Step 1: Factor the Denominator

The denominator $x^2 - x - 2$ can be factored as follows: \[ x^2 - x - 2 = (x - 2)(x + 1) \]

Step 2: Partial Fraction Decomposition

We express the integrand $\frac{7x - 5}{(x - 2)(x + 1)}$ in terms of partial fractions: \[ \frac{7x - 5}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \] Multiplying through by the denominator $(x - 2)(x + 1)$ and solving for $A$ and $B$ gives us the coefficients needed for integration.

Step 3: Integrate Each Term

After finding $A$ and $B$, we integrate each term separately over the interval $[4, 5]$: \[ \int_{4}^{5} \frac{7x - 5}{(x - 2)(x + 1)} \, dx = -4 \log(5) - 3 \log(2) + 3 \log(3) + 4 \log(6) \]

Step 4: Evaluate the Integral

Evaluating the integral yields: \[ \text{Result} \approx 1.9457 \]