`int_(0)^(4)((y^(2)-4y+5)sin(y-2)dy)/([2y^(2)-8y+1])` is equal to

Options

A. `0`
B. `2`
C. `-2`
D. none of these

Explanation

To solve the integral

\[ I = \int_{0}^{4} \frac{(y^2 - 4y + 5) \sin(y - 2)}{2y^2 - 8y + 1} \, dy, \]

we will perform a substitution and analyze the symmetry of the function.

### Step 1: Substitute \( z = y - 2 \)

Let \( z = y - 2 \). Then, we have:

\[ y = z + 2 \quad \text{and} \quad dy = dz. \]

When \( y = 0 \), \( z = -2 \), and when \( y = 4 \), \( z = 2 \). Thus, the limits of integration change from \( 0 \) to \( 4 \) into \( -2 \) to \( 2 \).

### Step 2: Rewrite the integral in terms of \( z \)

Now we can rewrite the integral:

\[ I = \int_{-2}^{2} \frac{((z + 2)^2 - 4(z + 2) + 5) \sin(z)}{2(z + 2)^2 - 8(z + 2) + 1} \, dz. \]

### Step 3: Simplify the numerator and denominator

Calculating the numerator:

\[ (z + 2)^2 - 4(z + 2) + 5 = z^2 + 4z + 4 - 4z - 8 + 5 = z^2 + 1. \]

Calculating the denominator:

\[ 2(z + 2)^2 - 8(z + 2) + 1 = 2(z^2 + 4z + 4) - 8z - 16 + 1 = 2z^2 + 8z + 8 - 8z - 16 + 1 = 2z^2 - 7. \]

Thus, we can rewrite the integral as:

\[ I = \int_{-2}^{2} \frac{(z^2 + 1) \sin(z)}{2z^2 - 7} \, dz. \]

### Step 4: Analyze the symmetry of the integrand

Next, we will analyze the integrand:

\[ f(z) = \frac{(z^2 + 1) \sin(z)}{2z^2 - 7}. \]

Notice that \( f(-z) = \frac{((-z)^2 + 1) \sin(-z)}{2(-z)^2 - 7} = \frac{(z^2 + 1)(-\sin(z))}{2z^2 - 7} = -f(z) \).

### Step 5: Conclude the integral

Since \( f(z) \) is an odd function (i.e., \( f(-z) = -f(z) \)), the integral of an odd function over a symmetric interval around zero is zero:

\[ I = \int_{-2}^{2} f(z) \, dz = 0. \]

### Final Answer

Thus, the value of the integral is

\[ \boxed{0}. \]