1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# Write a value of $\int \frac{1}{1+2{e}^{x}}dx$.

Open in App
Solution

## $\mathrm{Let}I=\int \frac{dx}{1+2{e}^{x}}\phantom{\rule{0ex}{0ex}}\mathrm{Dividing}\mathrm{numerator}&\mathrm{denominator}\mathrm{by}{e}^{x}\phantom{\rule{0ex}{0ex}}⇒I=\int \frac{\frac{1}{{e}^{x}}dx}{\frac{1}{{e}^{x}}+2}\phantom{\rule{0ex}{0ex}}=\int \frac{{e}^{-x}dx}{{e}^{-x}+2}\phantom{\rule{0ex}{0ex}}\mathrm{Let}{e}^{-x}+2=t\phantom{\rule{0ex}{0ex}}⇒-{e}^{-x}dx=dt\phantom{\rule{0ex}{0ex}}⇒{e}^{-x}dx=-dt\phantom{\rule{0ex}{0ex}}\therefore I=-\int \frac{dt}{t}\phantom{\rule{0ex}{0ex}}=-\mathrm{log}\left|t\right|+C\phantom{\rule{0ex}{0ex}}=-\mathrm{log}\left|{e}^{-x}+2\right|+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{t}\mathit{=}{\mathit{e}}^{\mathit{-}\mathit{x}}\mathit{+}\mathit{2}\right)$

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Special Integrals - 2
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program