Problem 25 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int e^x \sqrt{1 + e^x} \, dx \)

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{2}{3}(1 + e^x)^{3/2} + C \).

Step 1: Identify the Substitution

In order to evaluate this integral, we can use substitution to simplify the expression. Notice that the expression under the square root and the exponent is related. Let us substitute: \[ u = 1 + e^x \] Then the derivative is: \[ du = e^x \, dx \] Thus, \( e^x \, dx = du \).

Step 2: Rewrite the Integral

Substitute \( u = 1 + e^x \) and \( e^x \, dx = du \) into the integral:\[ \int e^x \sqrt{1 + e^x} \, dx = \int \sqrt{u} \, du \] This results in a simpler integral that we can solve using standard integration techniques.

Step 3: Integrate the Simplified Integral

Now integrate \( \int \sqrt{u} \, du \). We can rewrite \( \sqrt{u} \) as \( u^{1/2} \):\[ \int u^{1/2} \ du = \frac{u^{3/2}}{3/2} + C = \frac{2}{3}u^{3/2} + C \] Where \( C \) is the constant of integration.

Step 4: Substitute Back the Original Variable

Replace \( u \) with the original expression \( 1 + e^x \) to express the solution in terms of \( x \):\[ \frac{2}{3}(1 + e^x)^{3/2} + C \] This is the solution to the integral in terms of the original variable \( x \).

Key Concepts

Substitution Method

When faced with a complicated integral, the substitution method can make the process much easier. The idea is to choose a new variable, often denoted as \( u \), to replace a part of the integral that repeats or has a non-trivial derivative.

• Identify a part of the integral whose differentiation appears elsewhere in the problem. In our case, \( 1 + e^x \) was simplified using \( u = 1 + e^x \).
• Find the differential \( du \). If \( u = 1 + e^x \), then \( du = e^x \, dx \).
• Replace the original variables with \( u \) and \( du \). This converts the integral into a simpler form. In our integral, \( \int e^x \sqrt{1 + e^x} \, dx = \int \sqrt{u} \, du \).

The substitution method transforms challenging problems into ones you can solve with basic techniques. It effectively reduces the complexity of the integration process.

Simplifying Integrals

Simplifying integrals is an essential skill for solving complex integrals. It often involves identifying patterns and reformulating the integral into a more computable form.

• Using substitution, we simplified \( \int e^x \sqrt{1 + e^x} \, dx \) to \( \int \sqrt{u} \, du \). This step required recognizing the connection between \( e^x \) and its coefficient \( 1 + e^x \).
• Write the new integral with the substituted variable. In our example, transforming \( \sqrt{u} \) into an exponent \( u^{1/2} \) made the integration straightforward.

In simplified form, integrals fit standard formats and become much easier to evaluate. This allows for quick application of integration rules, yielding the solution without extra effort.

Constant of Integration

When evaluating indefinite integrals, always remember to add the constant of integration, denoted as \( C \). This constant accounts for the family of curves that differ only by a vertical shift.

• Every indefinite integral has this constant because differentiation of a constant does not affect the derivative.
• Mathematically, if \( F(x) \) is an antiderivative of \( f(x) \), then the most general antiderivative of \( f(x) \) is \( F(x) + C \).

For the given integral, after substituting back \( u = 1 + e^x \), the final expression was \( \frac{2}{3}(1 + e^x)^{3/2} + C \). This shows that any antiderivative of \( e^x \sqrt{1 + e^x} \) differs by just a constant. Always include this constant unless conditions specify otherwise.