Mathematics

$$\displaystyle \int \frac{\mathrm{cosec} ^{2}x-2005}{\cos ^{2005}x}dx$$ is equal to


ANSWER

none of these


SOLUTION
$$\displaystyle \int \frac{\csc^{2}x-2005}{\cos ^{2005}x}dx$$

$$=\displaystyle \int (\cos ^{-2005}x)\csc^{2}xdx-2005\int \frac{dx}{\cos ^{2005}x}$$

$$=\displaystyle (\cos x)^{-2005}(-\cot x)-\int (-2005)(\cos x)^{-2006}(-\sin x)(-\cot x)dx-2005\int \frac{dx}{\cos ^{2005}x}$$ ...... [Using integration by parts]

$$=-\displaystyle \frac{\cot x}{(\cos x)^{2005}}+C$$
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Single Correct Hard Published on 17th 09, 2020
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