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Integration by Parts

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Question

If $\int \frac{s e c^{2} x - 2010}{s i n^{2010} x} d x = \frac{P (x)}{(s i n x)^{2010}} + C$

A

$P (\frac{π}{4}) = \frac{1}{\sqrt{2}}$

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B

$P (\frac{π}{3}) = \sqrt{3}$

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C

$P (\frac{π}{2}) = 1$

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D

$P (\frac{π}{8}) + P (\frac{π}{4}) = \sqrt{2}$

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Solution

The correct option is B
$P (\frac{π}{3}) = \sqrt{3}$

$\int \frac{s e c^{2} x - 2010}{s i n^{2010} x} d x$
$= \int \frac{s e c^{2} x}{s i n^{2010} x} d x - \int \frac{2010}{s i n^{2010} x} d x$
$= \frac{t a n x}{(s i n x)^{2010}} + 2010 \int \frac{t a n x c o s x}{(s i n x)^{2011}} d x - \int \frac{2010}{s i n^{2010} x} d x + C$ (Applying Integration by parts in the first expression)
$= \frac{t a n x}{(s i n x)^{2010}} + \int \frac{2010}{(s i n x)^{2010}} d x - \int \frac{2010}{s i n^{2010} x} d x + C$
$= \frac{t a n x}{(s i n x)^{2010}} + C$
$∴ P (\frac{π}{3}) = t a n \frac{π}{3} = \sqrt{3}$

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