# $\int \frac{\sec ^{2}x-2010}{\sin ^{2010}x}dx=\frac{P(x)}{\sin ^{2010}x}+C\: \: value \: \: of\: \: P( \frac{\Pi}{3})\: \: is$ Option 1) 0 Option 2) $\frac{1}{\sqrt{3}}$ Option 3) $\sqrt3$ Option 4) none

G gaurav

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

$= \int{{{\sec }^2}x{{\left( {\sin x} \right)}^{ - 2010}} - 2010\int {\frac{1}{{{{\left( {\sin x} \right)}^{2010}}}}dx = {I_1} - {I_2}}}$

Applying by parts on I1, we get

${I_1} = \frac{{\tan x}}{{{{\left( {\sin x} \right)}^{2010}}}} + 2010\int {\frac{{\tan x\,\cos x}}{{{{\left( {\sin x} \right)}^{2011}}}}dx = \frac{{\tan x}}{{{{\left( {\sin x} \right)}^{2010}}}} + 2010\int {\frac{{dx}}{{{{\left( {\sin x} \right)}^{2010}}}}} }$

$\Rightarrow I={{I}_{1}}-{{I}_{2}}=\frac{\tan x}{{{\left( \sin x \right)}^{2010}}}=\frac{P\left( x \right)}{{{\left( \sin x \right)}^{2010}}}$

$P\left( \frac{\pi }{3} \right)=\tan \frac{\pi }{3}=\sqrt{3}$

Option 1)

0

Option 2)

$\frac{1}{\sqrt{3}}$

Option 3)

$\sqrt3$

Option 4)

none

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