The value of the integral `I=int_(0)^((pi)/(2))(cosx-sinx)/(10-x^(2)+(pix)/(2))dx` is equal to

Options

A. `(pi)/(2)`
B. `pi`
C. 0
D. `4pi`

Answer

To solve the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos x - \sin x}{10 - x^2 + \frac{\pi}{2} x} \, dx, \]

we can utilize a property of definite integrals. The property states that if \( f(x) \) is a function defined on the interval \([a, b]\), then:

\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx. \]

### Step 1: Apply the property

In our case, \( a = 0 \) and \( b = \frac{\pi}{2} \). Therefore, we can rewrite the integral as:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos(\frac{\pi}{2} - x) - \sin(\frac{\pi}{2} - x)}{10 - (\frac{\pi}{2} - x)^2 + \frac{\pi}{2}(\frac{\pi}{2} - x)} \, dx. \]

### Step 2: Simplify the numerator

Using the trigonometric identities, we have:

\[ \cos\left(\frac{\pi}{2} - x\right) = \sin x \quad \text{and} \quad \sin\left(\frac{\pi}{2} - x\right) = \cos x. \]

Thus, the numerator becomes:

\[ \sin x - \cos x. \]

### Step 3: Simplify the denominator

Now, let's simplify the denominator:

\[ 10 - \left(\frac{\pi}{2} - x\right)^2 + \frac{\pi}{2}\left(\frac{\pi}{2} - x\right). \]

Calculating \(\left(\frac{\pi}{2} - x\right)^2\):

\[ \left(\frac{\pi}{2} - x\right)^2 = \frac{\pi^2}{4} - \pi x + x^2. \]

Substituting this back into the denominator gives:

\[ 10 - \left(\frac{\pi^2}{4} - \pi x + x^2\right) + \frac{\pi^2}{4} - \frac{\pi}{2} x = 10 - \frac{\pi^2}{4} + \pi x - x^2 - \frac{\pi}{2} x. \]

Combining like terms results in:

\[ 10 - \frac{\pi^2}{4} + \left(\pi - \frac{\pi}{2}\right)x - x^2 = 10 - \frac{\pi^2}{4} + \frac{\pi}{2}x - x^2. \]

### Step 4: Write the new integral

Now we can write the integral as:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x - \cos x}{10 - \frac{\pi^2}{4} + \frac{\pi}{2}x - x^2} \, dx. \]

### Step 5: Combine the two integrals

Now we have:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos x - \sin x}{10 - x^2 + \frac{\pi}{2}x} \, dx \]
and
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin x - \cos x}{10 - \frac{\pi^2}{4} + \frac{\pi}{2}x - x^2} \, dx. \]

Adding these two expressions for \( I \):

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\cos x - \sin x + \sin x - \cos x}{10 - x^2 + \frac{\pi}{2}x} \right) \, dx = 0. \]

### Step 6: Solve for \( I \)

Thus, we have:

\[ 2I = 0 \implies I = 0. \]

### Final Answer

The value of the integral \( I \) is:

\[ \boxed{0}. \]