Misc 13 - Integrate ex / (1 + ex) (2 + ex) - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 13 Integrate the function 𝑒^π‘₯/((1 + 𝑒^π‘₯ )(2+γ€– 𝑒〗^π‘₯ ) ) ∫1▒𝑒^π‘₯/((1 + 𝑒^π‘₯ )(2+γ€– 𝑒〗^π‘₯ ) ) Let t = 𝑒^π‘₯ 𝑑𝑑/𝑑π‘₯ = 𝑒^π‘₯ dt = 𝑒^π‘₯ 𝑑π‘₯ Substituting ∫1▒𝑒^π‘₯/((1 + 𝑒^π‘₯ )(2+γ€– 𝑒〗^π‘₯ ) ) = ∫1▒𝑑𝑑/((1 + 𝑑) (2 + 𝑑)) We can write, 1/((1 + 𝑑)(2 + 𝑑))=A/(1 + 𝑑)+B/(2 + 𝑑) 1/((𝑑 + 1) (𝑑 + 2))=(A(t + 2) + B(t + 1))/((𝑑 + 1) (𝑑 + 2)) Cancelling denominators 1 = (t + 2)A + (t + 1)B Putting in (1) ∫1β–’1/((1 + 𝑑)(2 + 𝑑)) 𝑑𝑑=∫1β–’1/(𝑑 + 1)βˆ’1/(𝑑 + 2) 𝑑𝑑 = log |1+𝑑|βˆ’π‘™π‘œπ‘”|2+𝑑|+ C = log |(1 + 𝑑)/(2 + 𝑑)|+ C Putting t = 𝑒^π‘₯ = log ((𝟏 + 𝒆^𝒙)/(𝟐 + 𝒆^𝒙 ))+ C

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