$\large \displaystyle\int \dfrac{\sec^2(x) - 2010}{\sin^{2010}(x)}\, dx = \dfrac{P(x)}{(\sin(x))^{2010}} + C$

If the equation above is true for some function $P(x)$ with $P ( \frac{ \pi}{4} ) = \frac{1}{2^{1005} }$ , where $C$ denotes the arbitrary constant, then find the value of $P\left(\dfrac{\pi}{3}\right)$ .

$\dfrac{1}{\sqrt3}$
$\sqrt3$
1
0
$3\sqrt3$

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its simple

$\large{\int \frac{\sec^2 x}{(\sin x)^{2010}}dx-\int \frac{2010}{(\sin x)^{2010}}dx}$

In the former integral apply integration by parts considering $\frac{1}{(\sin x)^{2010}}$ as first function and $\sec^2 x$ as second

$\large{\frac{\tan x}{(\sin x)^{2010}}+\int {\frac{2010}{(\sin x)^{2010}}}dx-\int {\frac{2010}{(\sin x)^{2010}}}dx}$

so thus $P(x)=\tan (x)$