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Asked by vasoochaudhary | 15th Sep, 2014, 12:56: PM

Expert Answer:

integral fraction numerator sin space straight x plus cos space straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx equals integral fraction numerator sin space straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx plus integral fraction numerator cos space straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx equals integral fraction numerator sin space straight x over denominator 1 minus 2 space sin squared xcos squared straight x end fraction dx plus integral fraction numerator cos space straight x over denominator 1 minus 2 space sin squared xcos squared straight x end fraction dx space space space space left square bracket sin to the power of 4 straight x plus cos to the power of 4 straight x equals left parenthesis sin squared straight x plus cos squared straight x right parenthesis squared minus 2 sin squared xcos squared straight x equals 1 minus 2 sin squared xcos squared straight x right square bracket equals integral fraction numerator sin space straight x over denominator 1 minus 2 left parenthesis 1 minus cos squared straight x right parenthesis cos squared straight x end fraction dx plus integral fraction numerator cos space straight x over denominator 1 minus 2 space sin squared straight x left parenthesis 1 minus sin squared straight x right parenthesis end fraction dx let space straight t equals cos space straight x comma space straight p equals sin space straight x equals straight I subscript 1 plus straight I subscript 2............. left parenthesis straight i right parenthesis Solving space straight I subscript 2 equals integral fraction numerator cos space straight x over denominator 1 minus 2 space sin squared straight x left parenthesis 1 minus sin squared straight x right parenthesis end fraction dx straight p equals sin space straight x rightwards double arrow dp equals cos space straight x. dx equals integral fraction numerator dp over denominator 1 minus 2 straight p squared left parenthesis 1 minus straight p squared right parenthesis end fraction equals integral fraction numerator dp over denominator 1 minus 2 straight p squared plus 2 straight p to the power of 4 end fraction equals integral fraction numerator dp over denominator 1 minus 2 straight p squared plus 2 straight p to the power of 4 end fraction equals integral fraction numerator begin display style 1 over straight p squared end style dp over denominator begin display style 1 over straight p squared end style minus 2 plus 2 straight p squared end fraction left square bracket Dividing space Numerator space and space Denominator space by space straight p squared right square bracket equals integral fraction numerator begin display style open parentheses square root of 2 plus 1 over straight p squared close parentheses minus square root of 2 end style dp over denominator begin display style open parentheses straight p square root of 2 minus 1 over straight p close parentheses squared plus left parenthesis 2 square root of 2 minus 2 right parenthesis end style end fraction equals integral fraction numerator begin display style open parentheses square root of 2 plus 1 over straight p squared close parentheses end style dp over denominator open parentheses straight p square root of 2 minus 1 over straight p close parentheses squared plus open parentheses square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root close parentheses squared end fraction minus integral fraction numerator begin display style square root of 2 end style dp over denominator begin display style open parentheses straight p square root of 2 minus 1 over straight p close parentheses squared plus left parenthesis square root of 2 square root of 2 minus 2 end root right parenthesis squared end style end fraction let space open parentheses straight p square root of 2 plus 1 over straight p close parentheses equals straight m rightwards double arrow open parentheses square root of 2 plus 1 over straight p squared close parentheses dp equals dm equals integral fraction numerator dm over denominator straight m squared plus open parentheses square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root close parentheses squared end fraction minus integral fraction numerator begin display style square root of 2 end style dp over denominator begin display style open parentheses straight p square root of 2 minus 1 over straight p close parentheses squared plus left parenthesis square root of 2 square root of 2 minus 2 end root right parenthesis squared end style end fraction equals tan to the power of minus 1 end exponent open parentheses fraction numerator straight m over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator straight p square root of 2 minus 1 over straight p over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses Substituing space straight m space we space get equals tan to the power of minus 1 end exponent open parentheses fraction numerator straight p square root of 2 minus 1 over straight p over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator straight p square root of 2 minus 1 over straight p over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses Substituing space straight p space we space get comma equals tan to the power of minus 1 end exponent open parentheses fraction numerator sin space straight x square root of 2 minus fraction numerator 1 over denominator sin space straight x end fraction over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator sin space straight x square root of 2 minus fraction numerator 1 over denominator sin space straight x end fraction over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses Similarly comma space solving space straight I subscript 1 space we space get comma straight I subscript 1 equals integral fraction numerator sin space straight x over denominator 1 minus 2 left parenthesis 1 minus cos squared straight x right parenthesis cos squared straight x end fraction dx equals minus integral fraction numerator minus sin space straight x over denominator 1 minus 2 cos squared straight x plus 2 cos to the power of 4 straight x end fraction dx equals minus integral fraction numerator minus sin space straight x. dx over denominator 1 minus 2 cos squared straight x plus 2 cos to the power of 4 straight x end fraction equals minus integral fraction numerator dt over denominator 1 minus 2 straight t squared plus 2 straight t to the power of 4 end fraction equals tan to the power of minus 1 end exponent open parentheses fraction numerator cos space straight x square root of 2 minus fraction numerator 1 over denominator cos space straight x end fraction over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator cos space straight x square root of 2 minus fraction numerator 1 over denominator cos space straight x end fraction over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses Substituting space in space left parenthesis straight i right parenthesis space we space get comma equals tan to the power of minus 1 end exponent open parentheses fraction numerator cos space straight x square root of 2 minus fraction numerator 1 over denominator cos space straight x end fraction over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator cos space straight x square root of 2 minus fraction numerator 1 over denominator cos space straight x end fraction over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses plus tan to the power of minus 1 end exponent open parentheses fraction numerator sin space straight x square root of 2 minus fraction numerator 1 over denominator sin space straight x end fraction over denominator square root of left parenthesis 2 square root of 2 minus 2 right parenthesis end root end fraction close parentheses minus square root of 2 tan to the power of minus 1 end exponent open parentheses fraction numerator sin space straight x square root of 2 minus fraction numerator 1 over denominator sin space straight x end fraction over denominator square root of 2 square root of 2 minus 2 end root end fraction close parentheses

Answered by Prasenjit Paul | 16th Sep, 2014, 11:15: AM