Problem 18 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{e^{x}}{\left(1-e^{2 x}\right)^{3 / 2}} d x $$

Step-by-Step Solution

Answer

Integral of \(\int \frac{e^{x}}{(1-e^{2x})^{3 / 2}} dx\) is \(-\frac{e^{x}}{\sqrt{1-e^{2x}}}\).

Step 1: Identify Potential Substitutions

First, take notice of the outer expression \((1-e^{2x})\). It is advisable to translate it into a more user-friendly term. For this purpose, one can apply substitution in the following manner: Let \(u = e^{x}\). Then \(e^{2x} = u^{2}\), and the expression simplifies into \((1-u^{2})\). Potentially, substitution can help to simplify further calculation steps.

Step 2: Differential Substitution

Next, compute the differential of \(u\) i.e. \(du\). It is given as \(du = e^{x} dx\). Now, replace \(e^{x}dx\) in the integral with \(du\), the expression then simplifies to \(\int \frac{1}{(1-u^{2})^{3 / 2}} du\).

Step 3: Use Integration Tables

This simpler form falls under a standard form in integration tables. By checking in the table of integrals, we can find the integral of the function \(\frac{1}{(1-u^{2})^{3 / 2}}\) is \(-\frac{u}{\sqrt{1-u^{2}}}\).

Step 4: Back Substitute

Finally, we need to substitute \(u\) back into the obtained integral. Since \(u = e^{x}\), our final solution becomes \(-\frac{e^{x}}{\sqrt{1-e^{2x}}}\).