Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{3 e^{2 x}}{\sqrt{1-e^{2 x}}} d x $$1

Step-by-Step Solution

Answer

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

Step 1: Identify the Substitution

First, we notice that the expression under the square root, \( 1 - e^{2x} \), can be simplified using a substitution. Let's set \( u = 1 - e^{2x} \). Then, we'll differentiate \( u \) with respect to \( x \) to find \( du \).

Step 2: Differentiate the Substitution

Differentiating \( u = 1 - e^{2x} \) with respect to \( x \) gives us \( \frac{du}{dx} = -2e^{2x} \). Therefore, \( du = -2e^{2x} dx \), or \( e^{2x} dx = -\frac{1}{2} du \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) and \( e^{2x} dx \) into the integral. The integral becomes:\[\int \frac{3}{\sqrt{u}} \cdot \left(-\frac{1}{2} du\right)\]or simplified as:\[-\frac{3}{2} \int u^{-1/2} du\]

Step 4: Integrate the Simplified Expression

Now integrate \(-\frac{3}{2} \int u^{-1/2} du\). Recall that the integral \( \int u^{n} du = \frac{u^{n+1}}{n+1} + C \). Therefore, we get:\[-\frac{3}{2} \cdot \left( 2u^{1/2} \right) + C = -3 \sqrt{u} + C\]

Step 5: Substitute Back the Original Variable

Replace \( u \) with \( 1 - e^{2x} \) to revert the substitution:\[-3 \sqrt{1 - e^{2x}} + C\]

Step 6: Finalize the Expression

The final solution is:\[-3 \sqrt{1 - e^{2x}} + C\]where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method in calculus is a technique used to simplify integration by introducing a new variable. This is particularly helpful when dealing with complex expressions. In our original exercise, the expression under the square root, \(1 - e^{2x}\), is quite challenging to integrate directly. So, we introduce a substitution: set \(u = 1 - e^{2x}\). This helps transform the integral into an easier form.

Once the substitution is made, we differentiate \(u\) with respect to \(x\) to determine \(du\). This involves calculating the derivative and rewriting the integral in terms of \(du\) rather than \(dx\). The key step here is recognizing which part of the expression can be replaced to simplify integration.

• Choose a substitution that simplifies the integral.
• Differentiate the new variable.
• Rewrite the integral in terms of the new variable.

The substitution transforms a complex function into something more manageable, eventually making the integration step straightforward.

Definite and Indefinite Integrals

Integration, a fundamental concept in calculus, comes in two forms: definite and indefinite integrals. An indefinite integral, like in our example, finds a general function whose derivative matches the original integrand. It includes a constant of integration, represented as \(C\).

In our case, after substitution, the expression becomes \(-\frac{3}{2} \int u^{-1/2} \, du\). Solving this indefinite integral provides us with a function plus the constant of integration: \(-3 \sqrt{u} + C\).

• Definite integrals calculate a specific numerical value over an interval.
• Indefinite integrals find a general antiderivative and include the constant of integration \(C\).

When solving indefinite integrals, always remember to include the constant \(C\) since it represents any constant value that differentiates into zero.

Differentiation

Differentiation is the process of finding the derivative of a function. In substitution method, it plays a crucial role in moving from one variable to another. When setting \(u = 1 - e^{2x}\), we differentiated \(u\) with respect to \(x\), arriving at \( \frac{du}{dx} = -2e^{2x} \).

This result was later used to substitute \(dx\) in terms of \(du\), which is essential for rewriting the integral. Understanding differentiation allows us to manipulate expressions precisely, making integration possible. Differentiation tells us how quickly a function is changing at any point, which is key in translating between the variables.

• Useful in altering expressions during integration.
• Determines how one variable relates to another.

By calculating the derivative, we gain the ability to swap \(dx\) with \(du\), ensuring our integral is entirely in terms of a single variable.

Constant of Integration

The constant of integration, represented as \(C\), appears in indefinite integrals. When we integrate a function, it may represent a family of functions, all differing by a constant. This constant is crucial because derivatives of any constant are zero, meaning multiple functions can share the same derivative but differ by a constant.

In our solution, after integrating and simplifying, we obtained \(-3 \sqrt{1 - e^{2x}} + C\). The \(C\) is vital in expressing the entire set of antiderivatives, not just one specific function. Without it, the solution would represent an incomplete family of functions.

• Represents a family of solutions in indefinite integrals.
• Ensures full representation of all potential antiderivatives.

Always include the constant \(C\) when solving indefinite integrals to capture all possible solutions that have the same derivative as the integrand.