-->

Integrals

Question
CBSEENMA12032570

Show that:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos space straight x over denominator sin space straight x space plus space cos space straight x end fraction dx


Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos space straight x over denominator sin space straight x plus space cos space straight x end fraction dx                     ...(1)
Then straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx space space space space open square brackets because 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 straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator cosx plus sinx end fraction                             ...(2)
Adding (1) and (2), we get.
                 2 straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript open parentheses fraction numerator cos space straight x over denominator sin space straight x space plus space cos space straight x space end fraction plus fraction numerator sin space straight x over denominator sin space straight x plus space cos space straight x end fraction close parentheses dx space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cosx plus sinx over denominator sinx plus cosx end fraction dx space equals integral subscript 0 superscript straight pi over 2 end superscript 1 space dx
                      equals space open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2.
therefore space space space space space space 2 space space straight I space space equals space space straight pi over 2 space space space space space space space space space rightwards double arrow space space space space straight I space equals space straight pi over 4
                      

Some More Questions From Integrals Chapter