Question

Show that:
$integral subscript 0 superscript straight pi space straight x space. space log space sinx space dx space equals space minus straight pi squared over 2 log space 2$

Solution

Let I = $integral subscript 0 superscript straight pi straight x space log space sinx space dx$ ...(1)
$therefore space space space space space space space space space straight I space equals space integral subscript 0 superscript straight pi left parenthesis straight pi minus straight x right parenthesis space log space sin left parenthesis straight pi minus straight x right parenthesis space dx$    $open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore$    $straight I space equals space integral subscript 0 superscript straight pi left parenthesis straight pi minus straight x right parenthesis space log space sinx space dx space equals space straight pi integral subscript 0 superscript straight pi log space sinx space dx space minus space integral subscript 0 superscript straight pi straight x space log space sinx space dx$
$therefore space space space straight I space equals space straight pi integral subscript 0 superscript straight pi log space sinx space dx space minus space 1$    $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
$therefore space space space space space 2 space space straight I space equals space straight pi space space integral subscript 0 superscript straight pi log space sinx space dx space equals space straight pi. space 2 space integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx$
$therefore space space space space 2 straight I space equals space 2 straight pi open parentheses negative straight pi over 2 log space 2 close parentheses$    $open square brackets because space space integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx space equals negative straight pi over 2 log space 2 close square brackets$
$therefore space space space straight I equals space minus straight pi squared over 2 log space 2$

For Daily Free Study Material Join wiredfaculty

Integrals

Show that:
$integral subscript 0 superscript straight pi space straight x space. space log space sinx space dx space equals space minus straight pi squared over 2 log space 2$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction