In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

Verified step by step guidance

1. Start by identifying a substitution to simplify the integral. Notice the expression inside the denominator: \(1 + e^{2t}\). Let’s set \(x = e^{t}\). This substitution will help rewrite the integral in terms of \(x\).
2. Since \(x = e^{t}\), then \(dx = e^{t} dt = x dt\), which implies \(dt = \frac{dx}{x}\). Also, update the limits of integration: when \(t = \ln(\frac{3}{4})\), \(x = e^{\ln(\frac{3}{4})} = \frac{3}{4}\); when \(t = \ln(\frac{4}{3})\), \(x = e^{\ln(\frac{4}{3})} = \frac{4}{3}\).
3. Rewrite the integral in terms of \(x\): the numerator \(e^{t} dt\) becomes \(x dt = dx\), and the denominator becomes \((1 + x^{2})^{3/2}\). So the integral is now \(\int_{3/4}^{4/3} \frac{dx}{(1 + x^{2})^{3/2}}\).
4. Next, use a trigonometric substitution to handle the integral involving \(1 + x^{2}\). Since \(1 + x^{2}\) resembles the identity \(1 + \tan^{2}(\theta) = \sec^{2}(\theta)\), let’s set \(x = \tan(\theta)\). Then, \(dx = \sec^{2}(\theta) d\theta\).
5. Substitute into the integral: the denominator becomes \((1 + \tan^{2}(\theta))^{3/2} = (\sec^{2}(\theta))^{3/2} = \sec^{3}(\theta)\), and the numerator \(dx = \sec^{2}(\theta) d\theta\). The integral simplifies to \(\int \frac{\sec^{2}(\theta) d\theta}{\sec^{3}(\theta)} = \int \cos(\theta) d\theta\). Also, update the limits of integration by converting \(x = \tan(\theta)\) back to \(\theta\) using \(\theta = \arctan(x)\).

Key Concepts

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution u = g(t) such that the integral's expression becomes easier to integrate. This technique is essential when the integral contains composite functions or complicated expressions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. It is particularly useful for integrals containing expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function, the integral often reduces to a basic trigonometric integral.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specific points, defined by limits of integration. When performing substitution, it is important to adjust the limits according to the new variable or revert to the original variable after integration. Proper handling of limits ensures the correct evaluation of the integral's numerical value.