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Question: If \(\int_{}^{}\frac{\cos ec^{2}x - 2005}{\cos^{2005}x}\)dx = – \(\frac{A(x)}{(B(x))^{2005}}\)+ c, t...

If ∫cos⁡ec2x−2005cos⁡2005x\int_{}^{}\frac{\cos ec^{2}x - 2005}{\cos^{2005}x}dx = – A(x)(B(x))2005\frac{A(x)}{(B(x))^{2005}}+ c, then number of solutions of the equation A(x)B(x)\frac{A(x)}{B(x)}={x} in [0, 2p] is (where {.} represents fractional part function)

A

0

B

1

C

2

D

3

Answer

0

Explanation

Solution

∫cos⁡ec2x−2005cos⁡2005x\int_{}^{}\frac{\cos ec^{2}x - 2005}{\cos^{2005}x}dx

= ∫cos⁡ec2x⋅(cos⁡x)−2005\int_{}^{}{\cos ec^{2}x \cdot (\cos x)^{- 2005}}dx – 2005∫1(cos⁡x)2005\int_{}^{}\frac{1}{(\cos x)^{2005}}dx = I1 – I2

Applying by parts on I1, we get ∫cos⁡ec2x−2005cos⁡2005x\int_{}^{}\frac{\cos ec^{2}x - 2005}{\cos^{2005}x}dx

= −cot⁡x(cos⁡x)2005+C- \frac { \cot x } { ( \cos x ) ^ { 2005 } } + C

\ A(x) = cotx and B(x) = cos x

̃ = cosec x = {x} for x ∈[0, 2p] the equation has no solution as clear from the graph