Evaluate the following integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosec space straight x space cotx over denominator 1 minus cosec squared straight x end fraction dx$

Solution

Let I = $integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosecx space cosx over denominator 1 plus cosec squared straight x end fraction dx$
Put cosec x = y, $therefore space space space space minus cosecx space cot space straight x space dx space equals space dy$ $rightwards double arrow space space cosec space straight x space cotx space dx space equals space minus dy$
When $straight x equals straight pi over 6 comma space space space space space straight y space equals space cosec straight pi over 6 space equals space 2$
When $straight x equals straight pi over 2$ $straight y equals cosec straight pi over 2 space equals space 1$
$therefore space space space straight I space equals space space integral subscript 2 superscript 1 fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space minus open square brackets tan to the power of negative 1 end exponent space straight y close square brackets subscript 2 superscript 1 space equals space minus left parenthesis tan to the power of negative 1 end exponent 1 space minus space tan to the power of negative 1 end exponent 2 right parenthesis space equals space tan to the power of negative 1 end exponent 2 space minus space tan to the power of negative 1 end exponent 1$
$equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 minus 1 over denominator 1 plus 2 end fraction close parentheses space equals space tan to the power of negative 1 end exponent 1 third$

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Integrals

Evaluate the following integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosec space straight x space cotx over denominator 1 minus cosec squared straight x end fraction dx$

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