Question

If $\int \frac{{\sec }^{2}x-2010}{{\sin }^{2010}x}dx=\frac{P\left(x\right)}{(\sin x{)}^{2010}}+c$ , where $c$ is arbitrary constant then value of $P\left(\frac{\pi }{3}\right)$

• A. $0$
• B. $\frac{1}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $\frac{3\sqrt{3}}{2}$

Answer 1

The correct option is C $\sqrt{3}$

$\int \frac{{\sec }^{2}x-2010}{{\sin }^{2010}x}dx=\frac{P\left(x\right)}{(\sin x{)}^{2010}}+c$
Let $I=\int \frac{{\sec }^{2}x-2010}{{\sin }^{2010}x}dx$
$\Rightarrow I=\int {\sec }^{2}x.(\sin x{)}^{-2010}dx-2010\int (\sin x{)}^{-2010}dx$ ......(i)
Let $I_1=\int {\sec }^{2}x.(\sin x{)}^{-2010}dx$

Integrating by parts,
$\Rightarrow I_1=\frac{\tan x}{(\sin x{)}^{2010}}+2010\int \frac{\tan x\cos x}{(\sin x{)}^{2011}}dx+c$
$=\frac{\tan x}{(\sin x{)}^{2010}}+2010\int \frac{dx}{(\sin x{)}^{2010}}+c$
Putting this value in (i), we get
$\Rightarrow I=\frac{\tan x}{(\sin x{)}^{2010}}+c$
$\Rightarrow \frac{\tan x}{(\sin x{)}^{2010}}+c=\frac{P\left(x\right)}{(\sin x{)}^{2010}}+c$
$P\left(x\right)=\tan x$
$\therefore P\left(\frac{\pi }{3}\right)=\tan \frac{\pi }{3}=\sqrt{3}$