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Question Number 156600 by ZiYangLee last updated on 13/Oct/21

$$\int \frac{e^x}{e^{2x}+4}dx=?$$

Answered by gsk2684 last updated on 13/Oct/21

$$put{e}^x=t$$$$e^{x}dx=dt$$$$\int \frac{1}{t^2+2^2}dt$$$$=\frac{1}{2}{\tan }^{-1}\left(\frac{t}{2}\right)+c$$$$=\frac{1}{2}{\tan }^{-1}\left(\frac{e^x}{2}\right)+c$$

Answered by physicstutes last updated on 13/Oct/21

$$\mathrm{Another}\mathrm{cool}\mathrm{approach}.$$$$\int \frac{e^x}{e^{2x}+4}dx=\int \frac{1}{e^{2x}+4}{e}^{x}dx$$$$\mathrm{let}u=e^x\Rightarrow du=e^{x}dx$$$$\Rightarrow \int \frac{du}{u^2+4}du=\frac{1}{2}{\tan }^{-1}\left(\frac{u}{2}\right)+k$$$$=\frac{1}{2}{\tan }^{-1}\left(\frac{e^x}{2}\right)+k$$

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