PLz solve all these questions

Asked by 2330anurag | 29th Sep, 2015, 08:38: PM

Expert Answer:

C o n s i d e r space t h e space i n t e g r a l I equals integral fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction d x R e w r i t i n g space t h e space e x p r e s s i o n space fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction space a s comma fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction equals fraction numerator lambda begin display style fraction numerator d over denominator d x end fraction end style open parentheses 3 minus 2 x minus x squared close parentheses plus mu over denominator square root of 3 minus 2 x minus x squared end root end fraction rightwards double arrow fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction equals fraction numerator lambda open parentheses negative 2 minus 2 x close parentheses plus mu over denominator square root of 3 minus 2 x minus x squared end root end fraction rightwards double arrow fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction equals fraction numerator open parentheses negative 2 lambda close parentheses x plus open parentheses negative 2 lambda plus mu close parentheses over denominator square root of 3 minus 2 x minus x squared end root 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 s space w e space h a v e comma minus 2 lambda equals 3 space a n d space minus 2 lambda plus mu equals 1 rightwards double arrow lambda equals fraction numerator negative 3 over denominator 2 end fraction space a n d space minus 2 cross times open parentheses fraction numerator negative 3 over denominator 2 end fraction close parentheses plus mu equals 1 rightwards double arrow lambda equals fraction numerator negative 3 over denominator 2 end fraction space a n d space 3 plus mu equals 1 rightwards double arrow lambda equals fraction numerator negative 3 over denominator 2 end fraction space a n d space mu equals negative 2 T h u s comma fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction equals fraction numerator fraction numerator negative 3 over denominator 2 end fraction open parentheses negative 2 minus 2 x close parentheses minus 2 over denominator square root of 3 minus 2 x minus x squared end root end fraction T h u s comma I equals integral fraction numerator 3 x plus 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction d x space space equals integral open square brackets fraction numerator fraction numerator negative 3 over denominator 2 end fraction open parentheses negative 2 minus 2 x close parentheses minus 2 over denominator square root of 3 minus 2 x minus x squared end root end fraction close square brackets d x space space equals fraction numerator negative 3 over denominator 2 end fraction integral open square brackets fraction numerator open parentheses negative 2 minus 2 x close parentheses over denominator square root of 3 minus 2 x minus x squared end root end fraction close square brackets d x minus integral fraction numerator 2 over denominator square root of 3 minus 2 x minus x squared end root end fraction d x N o w space s u b s t i t u t i n g space t equals 3 minus 2 x minus x squared semicolon space d t equals open parentheses negative 2 minus 2 x close parentheses d x space 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 h a v e comma I equals fraction numerator negative 3 over denominator 2 end fraction integral fraction numerator d t over denominator square root of t end fraction minus 2 integral fraction numerator 1 over denominator square root of 3 minus 2 x minus x squared end root end fraction d x space equals fraction numerator negative 3 over denominator 2 end fraction cross times 2 square root of 3 minus 2 x minus x squared end root minus 2 integral fraction numerator 1 over denominator square root of 3 plus 1 minus 1 minus 2 x minus x squared end root end fraction d x space equals fraction numerator negative 3 over denominator 2 end fraction cross times 2 square root of 3 minus 2 x minus x squared end root minus 2 integral fraction numerator 1 over denominator square root of 4 minus open parentheses 1 plus 2 x plus x squared close parentheses end root end fraction d x space equals fraction numerator negative 3 over denominator 2 end fraction cross times 2 square root of 3 minus 2 x minus x squared end root minus 2 integral fraction numerator 1 over denominator square root of 2 squared minus open parentheses x plus 1 close parentheses squared end root end fraction d x space equals fraction numerator negative 3 over denominator 2 end fraction cross times 2 square root of 3 minus 2 x minus x squared end root minus 2 sin to the power of negative 1 end exponent fraction numerator x plus 1 over denominator 2 end fraction plus C space space space space space open square brackets because integral fraction numerator d x over denominator square root of a squared minus x squared end root end fraction equals s i n to the power of negative 1 end exponent x over a plus C close square brackets

Answered by Vimala Ramamurthy | 30th Sep, 2015, 10:18: AM