91Ó°ÊÓ

When dealing with integrals involving composite functions, substitution is a powerful technique. It simplifies integration by setting a new variable to make the integral more manageable. In the problem \( \int e^{x} \sqrt{1-e^{2x}} \, dx \), the integrand includes the expression \( 1 - e^{2x} \). This suggests that setting \( u = e^x \) could simplify our work. Therefore, \( du = e^x \, dx \), allowing us to replace \( e^x \, dx \) in our integral with \( du \). By making this substitution, you essentially replace a complex expression with a simpler variable, \( u \), making it easier to integrate. This step-by-step replacement process is fundamental in reducing calculative workload and navigating through the initial complexity of the integrand with ease.

Once the integral \( \int \sqrt{1-u^2} \, du \) is achieved, a new technique called trigonometric substitution is applied. This method is especially useful when dealing with expressions under a square root resembling the Pythagorean identity. The identity \( 1 = \sin^2\theta + \cos^2\theta \) suggests a substitution such as \( u = \sin \theta \), which implies \( du = \cos \theta \, d\theta \). Applying this to the integral \( \int \sqrt{1 - \sin^2 \theta} \, \cos \theta \, d\theta \), you simplify \( \sqrt{1 - \sin^2 \theta} \) to \( \cos \theta \). This simplification exploits the Pythagorean identity, transforming the integral into an easier form, \( \int \cos^2 \theta \, d\theta \). Trigonometric substitution is critical here, transforming complex algebraic forms into simpler trigonometric integrals.

The integral \( \int \cos^2 \theta \, d\theta \) can be resolved using another trigonometric tool – the double angle formula. By applying \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \), the integral splits into two separate, simpler integrals: \( \frac{1}{2} \int 1 \, d\theta + \frac{1}{2} \int \cos(2\theta) \, d\theta \). These integrals are straightforward to solve. The first integral evaluates to \( \frac{1}{2}\theta \), while the second becomes \( \frac{1}{4}\sin(2\theta) \) after integrating by recognizing the derivative. Combining these results gives you a solution in terms of \( \theta \). This formula not only simplifies calculations but also allows the bridging back to the original variable, resulting in the final expression in \( x \). Double angle formulas are invaluable in managing trig functions that otherwise complicate the integration process.