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Answer the following.
Evaluate: ∫ (eˣ)/(1 + e²ˣ) dx
Evaluate:
$\int \frac{e^x}{1 + e^{2x}} \, dx$
Answer
Answer
Integral: $\int \left(\frac{e^x}{1+e^{2x}}\right) dx$
Put $e^x = t$, so $e^x dx = dt$.
$\int \frac{dt}{1 + t^2} = \tan^{-1} t + C$.
Hence $\int \frac{e^x}{1+e^{2x}}dx = \tan^{-1}(e^x) + C$.
Final answer: $\tan^{-1}(e^x)$