Question
Download Solution PDF\(\rm\displaystyle\int \dfrac{1}{1+e^x}dx\) is equal to
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
\(\rm \displaystyle\int \dfrac{1}{x}dx= \log x + c\)
Calculation:
\(\rm \text{Let I}=\displaystyle\int \dfrac{1}{1+e^x}dx \\=\rm\displaystyle\int \dfrac{e^{-x}}{e^{-x}+1}dx \)
Assume e-x + 1 = t
Differenatiang with respect to x, we get
⇒ -e-x dx = dt
∴ e-x dx = -dt
\(\rm =\displaystyle\int \dfrac{-dt}{t}\\= -\log t + c\\=-\log (e^{-x}+1)+c\\=\log \left(\frac{1}{e^{-x}+1} \right )+c\\=\log \left(\frac{e^x}{1+e^x} \right )+c\)
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