Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{3 e^{2 x}}{\sqrt{1-e^{2 x}}} d x\)

Step-by-Step Solution

Answer

The integral is \(-3\sqrt{1 - e^{2x}} + C\)."

Step 1: Substitution Choice

Make the substitution to simplify the integral. Let \( u = 1 - e^{2x} \). Then, \( du = -2e^{2x} dx \). Solve for \( dx \) as \( dx = \frac{du}{-2e^{2x}} \).

Step 2: Adjust the Integral

Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{3 e^{2 x}}{\sqrt{1-e^{2 x}}} \cdot \frac{du}{-2e^{2x}} = -\frac{3}{2} \int \frac{1}{\sqrt{u}} \, du \].

Step 3: Integrate

The integral \( \int \frac{1}{\sqrt{u}} \, du \) corresponds to \( \int u^{-1/2} \, du \), which integrates to \( 2u^{1/2} + C \) where \( C \) is the integration constant. Thus, \(-\frac{3}{2} \times 2 \int u^{-1/2} \, du = -3\sqrt{u} + C_1\).

Step 4: Substitute Back the Original Variable

Replace \( u \) with \( 1 - e^{2x} \) to express the integral in terms of \( x \): \[ -3\sqrt{1 - e^{2x}} + C_1 \].

Key Concepts

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specific limits or bounds on the x-axis. Unlike indefinite integrals, definite integrals result in a numerical value rather than an expression with a constant.

To compute a definite integral, you must follow these steps:

• Identify the integrand, which is the function you are integrating.
• Determine the limits of integration, which specify the interval over which you integrate.
• Perform the integration of the function using techniques such as substitution or integration by parts.
• After integrating, evaluate the resulting expression at the upper and lower limits to find the definite integral.

Understanding definite integrals is crucial because they are applied in numerous calculations, from finding areas and volumes to solving problems in physics and engineering.

Indefinite Integrals

Indefinite integrals represent a family of functions that result when you integrate a function without specifying any limits. These integrals include a constant of integration, denoted as "C," which accounts for any constant that could be added to the function.

The process of finding an indefinite integral involves reversing differentiation. It requires:

• Identifying the integrand, the function to be integrated.
• Applying appropriate integration techniques, such as the power rule, substitution method, or trigonometric identities, to integrate the function.
• Always including a constant of integration, since differentiation of a constant results in zero.

Indefinite integrals are fundamental in mathematics because they help find antiderivatives, which are essential for solving differential equations and evaluating definite integrals.

Substitution Method

The substitution method is a powerful technique used to simplify complex integrals by transforming them into simpler forms. This method is particularly useful when dealing with integrals that include composite functions or chains of functions.

To use the substitution method:

• Choose a substitution variable, often denoted as "u," to replace a part of the integrand. This should simplify the integral significantly.
• Express "du" in terms of "dx" and the chosen substitution variable.
• Rewrite the integral in terms of "u" and "du." This often simplifies the integrand and makes the integral easier to solve.
• Integrate the new expression concerning "u."
• Finally, substitute back the original variable to express your result in terms of the original variable.

The substitution method effectively reduces the complexity of an integral, much like it does in the original exercise where it helps manage the complexity of integrating exponential functions.

Calculus Problem Solving

Calculus problem solving involves using integration and differentiation techniques to find solutions to various mathematical problems. This process includes deciding which technique to apply, solving the problem step by step, and checking the solution for accuracy.

Here are some essential steps in calculus problem solving:

• Understand the problem by identifying the quantities involved and what is being asked.
• Select appropriate formulas or theorems that apply to the specific problem scenario.
• Choose the right integration or differentiation technique based on the problem's requirements. Techniques like the substitution method, integration by parts, and partial fractions are frequently used.
• Work through the problem methodically, paying close attention to each step and calculation.
• Verify the solution by differentiating the results (in the case of integration) or checking that the solution satisfies any given conditions.
• Adjust and rework any part of the solution if errors are found at the verification stage.

Calculus problem solving is not just about finding answers; it's about understanding the underlying principles that guide us in everyday problem-solving scenarios.