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

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

dx

= dx

– 2005∫1(cos⁡x)2005\int_{}^{}\frac{1}{(\cos x)^{2005}}dx = I1 – I2

Applying by parts on I1, we get

dx = – cot⁡xcos⁡2005x\frac{\cot x}{\cos^{2005}x} + c

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

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