Solve:

Asked by joykphukan | 4th Jan, 2016, 11:03: AM

Expert Answer:

integral fraction numerator 1 over denominator sin x open parentheses 2 plus 3 cos x close parentheses end fraction d x equals integral fraction numerator sin x over denominator s i n squared x open parentheses 2 plus 3 c o s x close parentheses end fraction d x space left parenthesis m u l t i p l y i n g space b o t h space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space sin x right parenthesis S u b s t i t u t e space cos x equals t rightwards double arrow negative sin x d x equals d t T h e space i n t e g r a l space b e c o m e s minus integral fraction numerator d t over denominator open parentheses 1 minus t squared close parentheses open parentheses 2 plus 3 t close parentheses end fraction equals integral fraction numerator d t over denominator open parentheses t squared minus 1 close parentheses open parentheses 2 plus 3 t close parentheses end fraction U sin g space p a r t i a l space f r a c t i o n comma space fraction numerator 1 over denominator open parentheses t squared minus 1 close parentheses open parentheses 2 plus 3 t close parentheses end fraction equals fraction numerator A t plus B over denominator t squared minus 1 end fraction plus fraction numerator C over denominator 2 plus 3 t end fraction 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 space o f space n u m e r a t o r comma space w e space g e t space 3 A plus C equals 0 comma space 2 A plus 3 B equals 0 comma space 2 B minus C equals 1 S o l v i n g space t h e s e space e q u a t i o n s space w e space g e t space A equals 3 over 5 comma space B equals negative 2 over 5 comma space C equals negative 9 over 5 H e n c e comma space fraction numerator 1 over denominator open parentheses t squared minus 1 close parentheses open parentheses 2 plus 3 t close parentheses end fraction equals fraction numerator begin display style 3 over 5 end style t minus begin display style 2 over 5 end style over denominator t squared minus 1 end fraction plus fraction numerator negative begin display style 9 over 5 end style over denominator 2 plus 3 t end fraction equals open parentheses fraction numerator 3 over denominator 5 cross times 2 end fraction close parentheses fraction numerator 2 t over denominator t squared minus 1 end fraction minus 2 over 5 fraction numerator 1 over denominator t squared minus 1 end fraction minus 9 over 5 fraction numerator 1 over denominator 2 plus 3 t end fraction therefore integral fraction numerator d t over denominator open parentheses t squared minus 1 close parentheses open parentheses 2 plus 3 t close parentheses end fraction equals 3 over 10 integral fraction numerator 2 t over denominator t squared minus 1 end fraction d t minus 2 over 5 integral fraction numerator 1 over denominator t squared minus 1 end fraction d t minus 9 over 5 integral fraction numerator 1 over denominator 2 plus 3 t end fraction d t equals 3 over 10 cross times log open vertical bar t squared minus 1 close vertical bar minus 2 over 5 cross times 1 half l o g open vertical bar fraction numerator t minus 1 over denominator t plus 1 end fraction close vertical bar minus 9 over 5 cross times 1 third cross times log open parentheses 2 plus 3 t close parentheses plus C N o w space b a c k space s u b s t i t u t e space t equals cos x space t o space g e t space t h e space a n s w e r space i n space t e r m s space o f space apostrophe x apostrophe.

Answered by satyajit samal | 5th Jan, 2016, 10:55: AM