If - 2 x-2010-sin 210 x dx - P x -sin x 2010- CA-P -4-1-2B- P -3-3C- P -2-1D- P -8- P -4-2

The correct option is B P ( π/3 )=√(3) ∫sec^2x 2010/sin^2010 x dx =∫sec^2x /sin^2010xdx ∫2010/sin^2010xdx = tan x/(sin x)^2010 + 2010 ∫tan x cos x/(sin x)^ ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

If ∫ 2 x-2010...

Question

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

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

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$P (\frac{π}{3}) = \sqrt{3}$

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$P (\frac{π}{2}) = 1$

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

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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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If ∫sec2x−2010sin2010x dx=P(x)(sin x)2010+C

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