Problem 25 - Essential Calculus

Question

Find the integral. $$ \int e^{x} \sqrt{1-e^{2 x}} d x $$

Step-by-Step Solution

Answer

The integral of \( e^{x} \sqrt{1-e^{2 x}} \) with respect to \(x\) is \(e^x + C\).

Step 1: Substitute

Set the substitution \(u = e^x\). The differential of \(u\), \(du\), is equal to \(e^x dx\). Hence, the integral can be rewritten as \( \int \sqrt{1 - u^2} du \).

Step 2: Apply Trigonometric Substitution

In situations where the integral involves the square root of \(1 - x^2\), a trigonometric substitution of the form \(x = \sin(\theta)\) can be useful. Replacing \(u\) with \(\sin(\theta)\), the integral becomes \(\int \sqrt{1 - \sin^2(\theta)} d\theta = \int \cos(\theta) d\theta\).

Step 3: Simple Integration

Now, the integral of cos(\theta) is sin(\theta). However, note that in step 2, we made the substitution \(u = \sin(\theta)\). Thus, when integrating \(\cos(\theta)\), we end up with \(u\).

Step 4: Convert Back to Original Variable

Finally, since our original variable was \(x\), not \(u\) or \(\theta\), we have to substitute back in to obtain the answer in terms of \(x\). Remembering that \(u = e^x\), our final answer is \(e^x + C\), where \(C\) is the constant of integration.