how to integrate the following function
integral fraction numerator sin x over denominator sin 4 x end fraction d x

Asked by shaktiisgreat000 | 19th Nov, 2014, 07:39: PM

Expert Answer:

integral fraction numerator sin space x over denominator sin space 4 x end fraction d x equals integral fraction numerator sin space x over denominator 2 space sin space 2 x. cos space 2 x end fraction d x equals integral fraction numerator sin space x over denominator 4 sin space x. cos space x left parenthesis 1 minus 2 sin squared x right parenthesis end fraction d x equals 1 fourth integral fraction numerator d x over denominator cos space x left parenthesis 1 minus 2 sin squared x right parenthesis end fraction equals 1 fourth integral fraction numerator cos space x over denominator cos squared x left parenthesis 1 minus 2 sin squared x right parenthesis end fraction d x equals 1 fourth integral fraction numerator cos space x over denominator left parenthesis 1 minus sin squared x right parenthesis left parenthesis 1 minus 2 space sin squared x right parenthesis end fraction d x L e t space sin space x equals t therefore cos space x. d x equals d t equals 1 fourth integral fraction numerator d t over denominator left parenthesis 1 minus t squared right parenthesis left parenthesis 1 minus 2 t squared right parenthesis end fraction equals 1 fourth integral fraction numerator d t over denominator left parenthesis 1 plus t right parenthesis left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis end fraction fraction numerator 1 over denominator left parenthesis 1 plus t right parenthesis left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis end fraction equals fraction numerator A over denominator left parenthesis 1 plus t right parenthesis end fraction plus fraction numerator B over denominator left parenthesis 1 minus t right parenthesis end fraction plus fraction numerator C over denominator left parenthesis 1 plus t square root of 2 right parenthesis end fraction plus fraction numerator D over denominator left parenthesis 1 minus t square root of 2 right parenthesis end fraction rightwards double arrow 1 equals A left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis plus B left parenthesis 1 plus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis plus C left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis plus D left parenthesis 1 plus t right parenthesis left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis rightwards double arrow 1 equals A left parenthesis 1 minus t minus 2 t squared plus 2 t cubed right parenthesis plus B left parenthesis 1 plus t minus 2 t squared minus 2 t cubed right parenthesis plus C left parenthesis 1 minus t square root of 2 minus t squared plus t cubed square root of 2 right parenthesis plus D left parenthesis 1 plus t square root of 2 minus t squared minus t cubed square root of 2 right parenthesis rightwards double arrow 1 equals left parenthesis A plus B plus C plus D right parenthesis plus t left parenthesis minus A plus B plus C square root of 2 plus D square root of 2 right parenthesis plus t squared left parenthesis minus 2 A minus 2 B minus C minus D right parenthesis plus t cubed left parenthesis 2 A minus 2 B plus C square root of 2 minus D square root of 2 right parenthesis therefore A plus B plus C plus D equals 1 comma space minus A plus B plus C square root of 2 plus D square root of 2 equals 0 minus 2 A minus 2 B minus C minus D equals 0 2 A minus 2 B plus C square root of 2 minus D square root of 2 equals 0 S o l v i n g space t h e space a b o v e space e q u a t i o n s comma space w e space g e t A equals square root of 2 minus 1 half comma space B equals minus 1 half minus square root of 2 comma space C equals minus 1 comma space D equals 3 integral fraction numerator d x over denominator left parenthesis 1 plus t right parenthesis left parenthesis 1 minus t right parenthesis left parenthesis 1 plus t square root of 2 right parenthesis left parenthesis 1 minus t square root of 2 right parenthesis end fraction equals integral fraction numerator square root of 2 minus 1 half over denominator left parenthesis 1 plus t right parenthesis end fraction d t plus integral fraction numerator minus 1 half minus square root of 2 over denominator left parenthesis 1 minus t right parenthesis end fraction d t plus integral fraction numerator minus 1 over denominator left parenthesis 1 plus t square root of 2 right parenthesis end fraction d t plus integral fraction numerator 3 over denominator left parenthesis 1 minus t square root of 2 right parenthesis end fraction d t equals square root of 2 minus 1 half integral fraction numerator d t over denominator left parenthesis 1 plus t right parenthesis end fraction left parenthesis minus 1 half minus square root of 2 right parenthesis integral fraction numerator d t over denominator left parenthesis 1 minus t right parenthesis end fraction minus integral fraction numerator d t over denominator left parenthesis 1 plus t square root of 2 right parenthesis end fraction d t plus integral fraction numerator 3 over denominator left parenthesis 1 minus t square root of 2 right parenthesis end fraction d t equals open parentheses square root of 2 minus 1 half close parentheses log left parenthesis 1 plus t right parenthesis plus left parenthesis 1 half plus square root of 2 right parenthesis log space left parenthesis 1 minus t right parenthesis minus fraction numerator 1 over denominator square root of 2 end fraction log left parenthesis 1 plus t square root of 2 right parenthesis minus fraction numerator 3 over denominator square root of 2 end fraction log left parenthesis 1 minus t square root of 2 right parenthesis equals open parentheses square root of 2 minus 1 half close parentheses log left parenthesis 1 plus sin space x right parenthesis plus left parenthesis 1 half plus square root of 2 right parenthesis log space left parenthesis 1 minus sin space x right parenthesis minus fraction numerator 1 over denominator square root of 2 end fraction log left parenthesis 1 plus sin space x square root of 2 right parenthesis minus fraction numerator 3 over denominator square root of 2 end fraction log left parenthesis 1 minus sin space x square root of 2 right parenthesis

Answered by Prasenjit Paul | 21st Nov, 2014, 11:16: AM

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