Integrate sqrt (sin (x-a)/sin (x+a)) with respect to x.

Asked by Tanya raj | 16th Jan, 2016, 08:41: PM

Expert Answer:

begin mathsize 14px style L e t space I equals integral square root of fraction numerator sin left parenthesis x minus a right parenthesis over denominator sin left parenthesis x plus a right parenthesis end fraction end root d x equals integral square root of fraction numerator s i n left parenthesis x minus a right parenthesis over denominator s i n left parenthesis x plus a right parenthesis end fraction cross times fraction numerator sin left parenthesis x minus a right parenthesis over denominator sin left parenthesis x minus 1 right parenthesis end fraction end root d x equals integral fraction numerator sin left parenthesis x minus a right parenthesis over denominator square root of sin squared x minus sin squared a end root end fraction d x equals integral fraction numerator sin x cos a minus cos x sin a over denominator square root of sin squared x minus sin squared a end root end fraction d x equals cos a integral fraction numerator sin x over denominator square root of 1 minus cos squared x minus 1 plus cos squared a end root end fraction d x minus sin a integral fraction numerator cos x over denominator square root of s i n squared x minus s i n squared a end root end fraction d x equals cos a integral fraction numerator s i n x over denominator square root of c o s squared a minus c o s squared x end root end fraction d x minus sin a integral fraction numerator cos x over denominator square root of s i n squared x minus s i n squared a end root end fraction d x I n space t h e space f i r s t space i n t e g r a l comma space w e space p u t space cos x equals t comma space s o space t h a t space minus sin x d x equals d t space a n d space i n space t h e space s e c o n d space i n t e g r a l comma space w e space p u t sin x equals u comma space s o space t h a t space cos x d x equals d u I equals negative cos a integral fraction numerator d t over denominator square root of cos squared a minus t squared end root end fraction minus sin a integral fraction numerator d u over denominator square root of u squared minus sin squared a end root end fraction equals negative cos a sin to the power of negative 1 end exponent open parentheses fraction numerator t over denominator cos a end fraction close parentheses minus sin a log open vertical bar u plus square root of u squared minus sin squared a end root close vertical bar plus c equals negative cos a sin to the power of negative 1 end exponent open parentheses fraction numerator cos x over denominator cos a end fraction close parentheses minus sin a log open vertical bar sin x plus square root of sin squared x minus sin squared a end root close vertical bar plus c end style

Answered by Rashmi Khot | 18th Jan, 2016, 09:58: AM