SOLVE integral left curly bracket left parenthesis 1 plus x hat 4 right parenthesis divided by left parenthesis 1 plus x hat 6 right parenthesis right curly bracket d x

Asked by Devesh | 30th Nov, 2015, 07:51: PM

Expert Answer:

C o n s i d e r space t h e space i n t e g r a l comma I equals integral fraction numerator 1 plus x to the power of 4 over denominator 1 plus x to the power of 6 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 comma space fraction numerator 1 plus x to the power of 4 over denominator 1 plus x to the power of 6 end fraction space comma space w e space h a v e comma fraction numerator 1 plus x to the power of 4 over denominator 1 plus x to the power of 6 end fraction equals fraction numerator 1 plus x to the power of 4 over denominator 1 plus open parentheses x squared close parentheses cubed end fraction space space space space space space space space space space space equals fraction numerator 1 plus x to the power of 4 over denominator open parentheses 1 plus x squared close parentheses open parentheses 1 minus x squared plus x to the power of 4 close parentheses end fraction N o w space s u b s t i t u t e space x squared equals y space comma space w e space h a v e comma fraction numerator 1 plus y squared over denominator 1 plus y cubed end fraction equals fraction numerator 1 plus y squared over denominator open parentheses 1 plus y close parentheses open parentheses 1 minus y plus y squared close parentheses end fraction T h u s comma space u sin g space p a r t i a l space f r a c t i o n s comma space w e space h a v e comma fraction numerator 1 plus y squared over denominator open parentheses 1 plus y close parentheses open parentheses 1 minus y plus y squared close parentheses end fraction equals fraction numerator A over denominator 1 plus y end fraction plus fraction numerator B y plus C over denominator 1 minus y plus y squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 over denominator 3 open parentheses 1 plus x squared close parentheses end fraction plus fraction numerator 1 plus x squared over denominator 3 open parentheses 1 minus x squared plus x to the power of 4 close parentheses end fraction T h u s comma I equals integral open square brackets fraction numerator 2 over denominator 3 open parentheses 1 plus x squared close parentheses end fraction plus fraction numerator 1 plus x squared over denominator 3 open parentheses 1 minus x squared plus x to the power of 4 close parentheses end fraction close square brackets d x space equals integral fraction numerator 2 over denominator 3 open parentheses 1 plus x squared close parentheses end fraction d x plus integral fraction numerator 1 plus x squared over denominator 3 open parentheses 1 minus x squared plus x to the power of 4 close parentheses end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator 1 plus x squared over denominator 1 minus x squared plus x to the power of 4 end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator begin display style fraction numerator 1 plus x squared over denominator x squared end fraction end style over denominator begin display style fraction numerator 1 minus x squared plus x to the power of 4 over denominator x squared end fraction end style end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator begin display style 1 plus 1 over x squared end style over denominator begin display style 1 over x squared minus 1 plus x squared end style end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator begin display style 1 plus 1 over x squared end style over denominator begin display style x squared plus 1 over x squared minus 1 end style end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator begin display style 1 plus 1 over x squared end style over denominator begin display style x squared plus 1 over x squared minus 2 plus 2 minus 1 end style end fraction d x space equals 2 over 3 integral fraction numerator 1 over denominator 1 plus x squared end fraction d x plus 1 third integral fraction numerator begin display style 1 plus 1 over x squared end style over denominator begin display style open parentheses x minus 1 over x close parentheses squared plus 1 end style end fraction d x L e t space t equals x minus 1 over x semicolon space d t equals open parentheses 1 plus 1 over x squared close parentheses d x therefore I equals 2 over 3 tan to the power of negative 1 end exponent x plus 1 third integral fraction numerator d t over denominator t squared plus 1 end fraction space space space space space space equals 2 over 3 t a n to the power of negative 1 end exponent x plus 1 third t a n to the power of negative 1 end exponent t therefore I equals 2 over 3 t a n to the power of negative 1 end exponent x plus 1 third t a n to the power of negative 1 end exponent open parentheses x minus 1 over x close parentheses plus C o n s tan t

Answered by Vimala Ramamurthy | 1st Dec, 2015, 11:08: AM