+91-8076753736[email protected]
logo
logo
-->

Integrals

Question
CBSEENMA12032639
Wired Faculty App

Evaluate:
integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sin squared straight x end fraction dx.

Solution

Let I = integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sin squared straight x end fraction dx                                           ...(1)
therefore space space space space space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator straight pi minus straight x over denominator 1 plus sin squared left parenthesis straight pi minus straight x right parenthesis end fraction dx space space space space space rightwards double arrow space space space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator straight pi minus straight x over denominator 1 plus sin squared straight x end fraction dx
rightwards double arrow space space space space space space space straight I space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sin squared straight x end fraction dx space space minus integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sin squared straight x end fraction dx
therefore space space space space space straight I space equals space straight pi integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sin squared straight x end fraction dx minus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 integral subscript 0 superscript straight pi fraction numerator 1 over denominator 1 plus sin squared straight x end fraction dx space equals space 2 straight pi space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 1 plus sin squared straight x end fraction dx
               equals space 2 straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sin squared straight x plus cos squared straight x plus sin squared straight x end fraction dx space equals space 2 straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator 2 sin squared straight x plus cos squared straight x end fraction dx equals space 2 straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator begin display style fraction numerator 1 over denominator cos squared straight x end fraction end style over denominator begin display style fraction numerator 2 sin squared straight x over denominator cos squared straight x end fraction end style plus begin display style fraction numerator cos squared straight x over denominator cos squared straight x end fraction end style end fraction dx space equals space 2 straight pi integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sec squared straight x over denominator 2 space tan squared straight x plus 1 end fraction dx
Put tan x = t, ∴ sec 2 x dx = dt When x = 0, t = tan 0 = 0
When straight x space equals space straight pi over 2 comma space space straight t space equals space tan straight pi over 2 space rightwards arrow space space infinity
therefore space space space space space space space space space space straight I space equals space straight pi integral subscript 0 superscript infinity fraction numerator 1 over denominator 2 straight t squared plus 1 end fraction dt space equals space straight pi over 2 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus begin display style 1 half end style end fraction dt space equals space straight pi over 2 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus open parentheses begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses squared end fraction dt
                equals space straight pi over 2 cross times fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator straight t over denominator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction close parentheses close square brackets subscript 0 superscript infinity space equals space fraction numerator straight pi over denominator square root of 2 end fraction open square brackets tan to the power of negative 1 end exponent open parentheses square root of 2 straight t close parentheses close square brackets subscript 0 superscript infinity equals space fraction numerator straight pi over denominator square root of 2 end fraction left square bracket tan to the power of negative 1 end exponent infinity space minus space tan to the power of negative 1 end exponent 0 right square bracket space equals space fraction numerator straight pi over denominator square root of 2 end fraction open parentheses fraction numerator straight pi over denominator square root of 2 end fraction minus 0 close parentheses space equals space fraction numerator straight pi over denominator square root of 2 end fraction cross times straight pi over 2 space equals fraction numerator straight pi squared over denominator 2 square root of 2 end fraction

Some More Questions From Integrals Chapter

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation