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Sign of Trigonometric Ratios in Different Quadrants

∫cosec 2x+5dx...

Question

$\int cosec (2 x + 5) d x$ is equal to
(where $C$ is constant of integration)

\frac{ln | cosec (2 x + 5) + cot (2 x + 5) |}{2} + C

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\frac{ln | cosec (2 x + 5) + cot (2 x + 5) |}{5} + C

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- \frac{ln | cosec (2 x + 5) + cot (2 x + 5) |}{5} + C

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- \frac{ln | cosec (2 x + 5) + cot (2 x + 5) |}{2} + C

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Solution

The correct option is D $- \frac{ln | cosec (2 x + 5) + cot (2 x + 5) |}{2} + C$
$\int cosec (2 x + 5) d x = \frac{- ln | cosec (2 x + 5) + cot (2 x + 5) |}{2} + C$
$(∵ \int cosec (a x + b) d x = \frac{- ln | cosec (a x + b) + cot (a x + b) |}{a} + C, a \neq 0)$

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