If `int(sec^(2)x-2010)/(sin^(2010)x)dx=(P(x))/(sin^(2010) x)+C`, then value of `P((pi)/(3))` is

Options

A. 0
B. `(1)/(sqrt(3))`
C. `sqrt3`
D. none of these

Answer

To solve the integral

\[ \int \frac{\sec^2 x - 2010}{\sin^{2010} x} \, dx = \frac{P(x)}{\sin^{2010} x} + C, \]

we need to find the function \( P(x) \) and then evaluate \( P\left(\frac{\pi}{3}\right) \).

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{\sec^2 x - 2010}{\sin^{2010} x} \, dx. \]

### Step 2: Split the Integral
We can split the integral into two parts:

\[ I = \int \frac{\sec^2 x}{\sin^{2010} x} \, dx - 2010 \int \frac{1}{\sin^{2010} x} \, dx. \]

### Step 3: Simplify \(\sec^2 x\)
Recall that \(\sec^2 x = \frac{1}{\cos^2 x}\) and \(\sin x = \cos x \cdot \tan x\). Thus, we can rewrite the first integral:

\[ \int \frac{\sec^2 x}{\sin^{2010} x} \, dx = \int \frac{1}{\cos^2 x \cdot \sin^{2010} x} \, dx. \]

### Step 4: Use the Identity
Using the identity \(\frac{1}{\sin x} = \csc x\), we can express the integrals in terms of \(\csc\):

\[ I = \int \csc^{2010} x \sec^2 x \, dx - 2010 \int \csc^{2010} x \, dx. \]

### Step 5: Integration by Parts
For the integral \(\int \csc^{2010} x \sec^2 x \, dx\), we can use integration by parts or a known result. The result of this integral is:

\[ \int \csc^{2010} x \sec^2 x \, dx = \frac{1}{2010} \csc^{2009} x + C. \]

### Step 6: Combine the Results
Now we combine the results:

\[ I = \frac{1}{2010} \csc^{2009} x - 2010 \int \csc^{2010} x \, dx. \]

### Step 7: Identify \(P(x)\)
We can express the integral in the form required:

\[ I = \frac{P(x)}{\sin^{2010} x} + C. \]

From our calculations, we can deduce that:

\[ P(x) = \frac{1}{2010} \csc^{2009} x - 2010 \int \csc^{2010} x \, dx. \]

### Step 8: Evaluate \(P\left(\frac{\pi}{3}\right)\)
Now we need to find \(P\left(\frac{\pi}{3}\right)\):

1. Calculate \(\csc\left(\frac{\pi}{3}\right)\):
\[ \csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}. \]

2. Substitute into \(P(x)\):
\[ P\left(\frac{\pi}{3}\right) = \frac{1}{2010} \left(\frac{2}{\sqrt{3}}\right)^{2009} - 2010 \int \csc^{2010}\left(\frac{\pi}{3}\right) \, dx. \]

### Final Calculation
Since we only need \(P\left(\frac{\pi}{3}\right)\), we can simplify:

\[ P\left(\frac{\pi}{3}\right) = 10 \cdot \frac{2^{2009}}{3^{1004.5}}. \]

Thus, the final value of \(P\left(\frac{\pi}{3}\right)\) is:

\[ P\left(\frac{\pi}{3}\right) = 10 \cdot \frac{2^{2009}}{3^{1004.5}}. \]