Let I = ∫e^2xe^2x 2dx Put t = e^2x 2 , then dt = 2e^2xdx . So, I = 12∫dtt = 12log| t | + c = 12log| e^2x 2| + c ...

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Integration by Substitution

Solve ∫e2xe...

Question

Solve $\int \frac{e^{2 x}}{e^{2 x} - 2} d x$

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Solution

Let $I = \int \frac{e^{2 x}}{e^{2 x} - 2} d x$

Put $t = e^{2 x} - 2$ , then $d t = 2 e^{2 x} d x$ . So,

$I = \frac{1}{2} \int \frac{d t}{t}$

$= \frac{1}{2} log | t | + c$

$= \frac{1}{2} log ∣ ∣ e^{2 x} - 2 ∣ ∣ + c$

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