CBSE Class 12-science Answered

Solutions

Asked by alanpeter9611 | 13 Feb, 2019, 02:40: AM

Expert Answer

begin mathsize 12px style L e t space I space equals space integral fraction numerator e to the power of x d x over denominator open parentheses e to the power of x plus 2 close parentheses open parentheses e to the power of 2 x end exponent plus 4 close parentheses end fraction l e t space u s space s u b s t i t u t e space space e to the power of x space equals space u space comma space space e to the power of x d x space equals space d u I space equals space integral fraction numerator d u over denominator open parentheses u plus 2 close parentheses open parentheses u squared plus 4 close parentheses end fraction l e t space u s space r e s o l v e space t h e space i n t e g r a n d space i n t o space p a r t i a l space f r a c t i o n fraction numerator 1 over denominator open parentheses u plus 2 close parentheses open parentheses u squared plus 4 close parentheses end fraction space equals space fraction numerator A over denominator u plus 2 end fraction plus fraction numerator B u plus C over denominator u squared plus 4 end fraction h e n c e space A open parentheses u squared plus 4 close parentheses space plus space open parentheses B u plus C close parentheses open parentheses u plus 2 close parentheses space space equals space 1 space space space space space space space space space..................... left parenthesis 1 right parenthesis end style

By comparing coefficents of u² , u and constant term, we get equations involving A, B and C.

By solving these equations we get A = 1/8, B = -1/8 and C = 1/4

begin mathsize 12px style H e n c e space w e space w r i t e comma space space I space equals space 1 over 8 integral fraction numerator d u over denominator u plus 2 end fraction space minus 1 over 16 integral fraction numerator 2 u space d u over denominator u squared plus 4 end fraction space space plus space 1 fourth integral fraction numerator d u over denominator u squared plus 4 end fraction I space equals space 1 over 8 log fraction numerator u plus 2 over denominator square root of open parentheses u squared plus 4 close parentheses end root end fraction space plus space 1 over 8 tan to the power of negative 1 end exponent open parentheses u over 2 close parentheses space plus space C I space equals space 1 over 8 log fraction numerator e to the power of x plus 2 over denominator square root of open parentheses e to the power of 2 x end exponent plus 4 close parentheses end root end fraction space plus space 1 over 8 tan to the power of negative 1 end exponent open parentheses e to the power of x over 2 close parentheses space plus space C end style

Answered by Thiyagarajan K | 13 Feb, 2019, 12:29: PM

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