A=integral subscript 0 superscript straight pi sin x divided by left parenthesis sin x plus cos x right parenthesis d x
b=integral subscript 0 superscript straight pi sin x divided by left parenthesis sin x minus cos x right parenthesis d x
THEN A&B ARE-

Asked by Devesh | 4th Oct, 2015, 02:30: PM

Expert Answer:

C o n s i d e r space A equals integral subscript 0 superscript straight pi fraction numerator s i n x over denominator s i n x plus c o s x end fraction d x R e w r i t i n g space s i n x equals K open parentheses s i n x plus c o s x close parentheses plus L fraction numerator d over denominator d x end fraction open parentheses s i n x plus c o s x close parentheses rightwards double arrow space s i n x equals K open parentheses s i n x plus c o s x close parentheses plus L open parentheses c o s x minus s i n x close parentheses rightwards double arrow space s i n x equals K s i n x plus K c o s x plus L c o s x minus L s i n x rightwards double arrow space s i n x equals open parentheses K minus L close parentheses s i n x plus open parentheses K plus L close parentheses c o s x C o m p a r i n g space t h e space c o e f f i c i e n t s comma space w e space h a v e comma K minus L equals 1 space a n d space K plus L equals 0 rightwards double arrow 2 K equals 1 rightwards double arrow K equals 1 half space a n d space L equals negative 1 half T h u s comma A equals integral fraction numerator s i n x over denominator s i n x plus c o s x end fraction d x space space space space equals integral fraction numerator 1 half open parentheses s i n x plus c o s x close parentheses minus 1 half open parentheses c o s x minus s i n x close parentheses over denominator s i n x plus c o s x end fraction d x space space space space equals 1 half integral d x minus 1 half integral fraction numerator open parentheses c o s x minus s i n x close parentheses over denominator s i n x plus c o s x end fraction d x space space space space equals 1 half integral d x plus 1 half integral fraction numerator open parentheses s i n x minus c o s x close parentheses over denominator s i n x plus c o s x end fraction d x space space space space equals open parentheses x over 2 close parentheses plus 1 half integral fraction numerator open parentheses s i n x minus c o s x close parentheses over denominator s i n x plus c o s x end fraction d x  T h u s comma A equals open parentheses x over 2 close parentheses minus 1 half integral fraction numerator f apostrophe open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction d x comma space w h e r e space f open parentheses x close parentheses equals s i n x plus c o s x space space space equals x over 2 minus 1 half l n open parentheses f open parentheses x close parentheses close parentheses space space space equals x over 2 minus 1 half l n open parentheses s i n x plus c o s x close parentheses A t space t h e space p o i n t space fraction numerator 3 straight pi over denominator 4 end fraction comma space t h i s space i n t e g r a t i o n space i s space i m p r o p e r. H e n c e space w e space n e e d space t o space d i v i d e space i n t g e g r a l space i n space t h e space f o l l o w i n g space s u b i n t e r v a l s colon space open parentheses 0 comma fraction numerator 3 straight pi over denominator 4 end fraction close parentheses space a n d space open parentheses fraction numerator 3 straight pi over denominator 4 end fraction comma straight pi close parentheses H e n c e space w e space n e e d space t o space u s e space F u n d a m e n t a l space T h e o r e m space o f space C a l c u l u s. T h u s N o w space c o n s i d e r space t h e space i n t e g r a l integral subscript 0 superscript fraction numerator 3 straight pi over denominator 4 end fraction end superscript fraction numerator s i n x over denominator s i n x plus c o s x end fraction d x space space space equals stack l i m with x rightwards arrow fraction numerator 3 straight pi over denominator 4 end fraction below open square brackets x over 2 minus 1 half l n open parentheses s i n x plus c o s x close parentheses close square brackets minus open square brackets x over 2 minus 1 half l n open parentheses s i n x plus c o s x close parentheses close square brackets subscript x equals 0 end subscript equals infinity  S i m i l a r l y comma space B equals integral subscript 0 superscript straight pi fraction numerator s i n x over denominator s i n x minus c o s x end fraction d x S o l v i n g space i n space t h e space s a m e space m a n n e r comma space w e space h a v e T h u s comma B equals open parentheses x over 2 close parentheses plus 1 half integral fraction numerator f apostrophe open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction d x comma space w h e r e space f open parentheses x close parentheses equals s i n x minus c o s x space space space equals x over 2 plus 1 half l n open parentheses f open parentheses x close parentheses close parentheses space space space equals x over 2 plus 1 half l n open parentheses s i n x minus c o s x close parentheses A t space t h e space p o i n t space straight pi over 4 comma space t h i s space i n t e g r a t i o n space i s space i m p r o p e r. H e n c e space w e space n e e d space t o space d i v i d e space i n t g e g r a l space i n space t h e space f o l l o w i n g space s u b i n t e r v a l s colon space open parentheses 0 comma straight pi over 4 close parentheses space a n d space open parentheses straight pi over 4 comma straight pi close parentheses H e n c e space w e space n e e d space t o space u s e space F u n d a m e n t a l space T h e o r e m space o f space C a l c u l u s. T h u s N o w space c o n s i d e r space t h e space i n t e g r a l integral subscript 0 superscript straight pi over 4 end superscript fraction numerator s i n x over denominator s i n x minus c o s x end fraction d x space space space equals stack l i m with x rightwards arrow straight pi over 4 below open square brackets x over 2 plus 1 half l n open parentheses s i n x minus c o s x close parentheses close square brackets minus open square brackets x over 2 plus 1 half l n open parentheses s i n x minus c o s x close parentheses close square brackets subscript x equals 0 end subscript equals negative infinity T h u s comma space b o t h space t h e space i n t e g r a l s space a r e space d i v e r g e n t.

Answered by Vimala Ramamurthy | 4th Oct, 2015, 03:33: PM