The value of `int_(3)^(5)(x^(2))/(x^(2)-4)` dx is

Options

A. `2-"log"_(e)((15)/(7))`
B. `2+"log"_(e)((15)/(7))`
C. `2+4" log"_(e)3-4" log"_(e)7+4" log"_(e)5`
D. `2-tan^(-1)((15)/(7))`

Explanation

To solve the integral \( \int_{3}^{5} \frac{x^2}{x^2 - 4} \, dx \), we can follow these steps:

### Step 1: Simplify the Integrand
We start by rewriting the integrand:
\[ \frac{x^2}{x^2 - 4} = 1 + \frac{4}{x^2 - 4} \]
This is done by performing polynomial long division.

### Step 2: Set Up the Integral
Now we can express the integral as:
\[ \int_{3}^{5} \left( 1 + \frac{4}{x^2 - 4} \right) \, dx \]
This can be split into two separate integrals:
\[ \int_{3}^{5} 1 \, dx + \int_{3}^{5} \frac{4}{x^2 - 4} \, dx \]

### Step 3: Evaluate the First Integral
The first integral is straightforward:
\[ \int_{3}^{5} 1 \, dx = [x]_{3}^{5} = 5 - 3 = 2 \]

### Step 4: Evaluate the Second Integral
For the second integral, we can simplify it further:
\[ \int_{3}^{5} \frac{4}{x^2 - 4} \, dx = 4 \int_{3}^{5} \frac{1}{(x-2)(x+2)} \, dx \]
Using partial fraction decomposition, we can write:
\[ \frac{1}{(x-2)(x+2)} = \frac{A}{x-2} + \frac{B}{x+2} \]
Solving for \(A\) and \(B\), we find:
\[ 1 = A(x+2) + B(x-2) \]
Setting \(x = 2\) gives \(A = \frac{1}{4}\) and setting \(x = -2\) gives \(B = -\frac{1}{4}\). Thus:
\[ \frac{1}{(x-2)(x+2)} = \frac{1/4}{x-2} - \frac{1/4}{x+2} \]
So the integral becomes:
\[ 4 \int_{3}^{5} \left( \frac{1/4}{x-2} - \frac{1/4}{x+2} \right) \, dx = \int_{3}^{5} \left( \frac{1}{x-2} - \frac{1}{x+2} \right) \, dx \]

### Step 5: Evaluate the Integral
Now we can evaluate:
\[ \int_{3}^{5} \frac{1}{x-2} \, dx - \int_{3}^{5} \frac{1}{x+2} \, dx \]
Calculating these integrals gives:
\[ \left[ \log |x-2| \right]_{3}^{5} - \left[ \log |x+2| \right]_{3}^{5} \]
Calculating the limits:
\[ \left( \log(5-2) - \log(3-2) \right) - \left( \log(5+2) - \log(3+2) \right) = \left( \log(3) - \log(1) \right) - \left( \log(7) - \log(5) \right) \]
This simplifies to:
\[ \log(3) - \log(7) + \log(5) = \log\left(\frac{15}{7}\right) \]

### Step 6: Combine Results
Now we combine the results from both integrals:
\[ 2 + \log\left(\frac{15}{7}\right) \]

### Final Answer
Thus, the value of the integral \( \int_{3}^{5} \frac{x^2}{x^2 - 4} \, dx \) is:
\[ \boxed{2 + \log\left(\frac{15}{7}\right)} \]