(0 to 1)  (e^-x).dx / (1+e^x)

Asked by sunil2791 | 3rd Oct, 2017, 07:16: PM

Expert Answer:

begin mathsize 16px style straight I equals integral subscript 0 superscript 1 fraction numerator straight e to the power of negative straight x end exponent over denominator 1 plus straight e to the power of straight x end fraction dx space equals integral subscript 0 superscript 1 fraction numerator begin display style fraction numerator begin display style 1 end style over denominator straight e to the power of straight x end fraction end style over denominator begin display style 1 plus straight e to the power of straight x end style end fraction dx space
Put space straight e to the power of straight x space equals space straight u rightwards double arrow straight e to the power of straight x space dx equals space du rightwards double arrow dx space equals space du over straight u
For space straight x space equals space 0 space rightwards double arrow straight u space equals space 1
For space straight x space equals space 1 space rightwards double arrow straight u space equals space straight e
straight I space equals space integral subscript 1 superscript straight e space fraction numerator begin display style 1 over straight u end style over denominator 1 plus straight u end fraction cross times fraction numerator begin display style du end style over denominator begin display style straight u end style end fraction
straight I space equals space integral subscript 1 superscript straight e fraction numerator du over denominator straight u squared space open parentheses 1 plus straight u close parentheses end fraction
Using space partial space fraction comma space you space can space solve space it space further.
end style

Answered by Sneha shidid | 28th Dec, 2017, 10:16: AM