Question

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx space cosecx end fraction space equals space straight pi squared over 4$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight x space tanx over denominator secx space cosecx end fraction dx$ ...(1)
$therefore space space space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator left parenthesis straight pi minus straight x right parenthesis space tan left parenthesis straight pi minus straight x right parenthesis over denominator sec left parenthesis straight pi minus straight x right parenthesis space cosec left parenthesis straight pi minus straight x right parenthesis end fraction dx$ $open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space space space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator left parenthesis straight pi minus straight x right parenthesis space left parenthesis negative tan space straight x right parenthesis over denominator left parenthesis negative sec space straight x right parenthesis space cosecx end fraction dx$
$straight I space equals straight pi integral subscript 0 superscript straight pi fraction numerator tan space straight x over denominator sec space straight x space cosec space straight x end fraction dx space space minus space integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator sec space straight x space cosec space straight x end fraction dx$
$rightwards double arrow space space space space straight I space equals space straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style sinx over cosx end style over denominator begin display style 1 over cosx end style. begin display style 1 over sinx end style end fraction dx minus 1$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$
$rightwards double arrow space space space space 2 space straight I space equals space straight pi integral subscript 0 superscript straight pi over 2 end superscript sin squared space xdx space equals space straight pi over 2 integral subscript 0 superscript straight pi over 2 end superscript 2 space sin squared straight x space dx rightwards double arrow space space space space 2 space straight I space equals space straight pi over 2 integral subscript 0 superscript straight pi left parenthesis 1 minus cos space 2 straight x right parenthesis space dx space equals space straight pi over 2 open square brackets straight x minus fraction numerator sin space 2 straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi therefore space space space space space 2 space straight I space equals space straight pi over 2 open square brackets open parentheses straight pi minus 1 half sin space 2 straight pi close parentheses space minus space open parentheses 0 minus 1 half sin space 0 close parentheses close square brackets therefore space space space 2 space straight I space equals space straight pi over 2 open square brackets open parentheses straight pi over 2 minus 1 half cross times 0 close parentheses space minus space left parenthesis 0 minus 0 right parenthesis close square brackets space space space space rightwards double arrow space space space 2 space straight I space equals space straight pi squared over 2 space space rightwards double arrow space space straight I space equals straight pi squared over 4$
$therefore space space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator left parenthesis straight pi minus straight x right parenthesis space left parenthesis negative tan space straight x right parenthesis over denominator left parenthesis negative sec space straight x right parenthesis space cosecx end fraction dx$

For Daily Free Study Material Join wiredfaculty

Integrals

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx space cosecx end fraction space equals space straight pi squared over 4$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction