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Magazine Chapter 5: Problem 19
Evaluate the indefinite integral. $$\int e^{x} \sqrt{1+e^{x}} d x$$
Short Answer
Expert verified
The indefinite integral is \( \frac{2}{3} (1 + e^x)^{3/2} + C \).
Step by step solution
01
Choose a Substitution
To simplify the integration, we notice the form of the integrand and choose a substitution. Let \( u = 1 + e^x \). This implies \( du = e^x \, dx \). This substitution will simplify the square root and the exponential.
02
Substitute in terms of u
Replace \( e^x \, dx \) with \( du \). Thus, the integral becomes \( \int \sqrt{u} \, du \). We have eliminated the exponential and the square root is now in a simpler form.
03
Integrate with Respect to u
The integral \( \int \sqrt{u} \, du \) can be rewritten as \( \int u^{1/2} \, du \). This is a standard power rule integral. The integral of \( u^{1/2} \) is \( \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \).
04
Re-substitute back to x
Substitute \( u = 1 + e^x \) back into the expression \( \frac{2}{3} u^{3/2} \). Thus, the integral becomes \( \frac{2}{3} (1 + e^x)^{3/2} \).
05
Add the Constant of Integration
To make it the indefinite integral, add the constant of integration \( C \). Therefore, the final answer is \( \frac{2}{3} (1 + e^x)^{3/2} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a crucial technique in calculus to simplify the process of evaluating integrals. When you encounter an integral that seems complicated, like the one involving the expression \( e^x \sqrt{1+e^x} \), identifying a term to substitute can transform it into a more manageable problem.
Here's how the process unfolds:
Here's how the process unfolds:
- First, pick a substitution variable, such as \( u \), to replace part of the integral that will simplify the form. In our example, we chose \( u = 1 + e^x \), reflecting the structure inside the square root.
- Next, calculate the differential of \( u \), denoted \( du \). Using derivative rules, we find \( du = e^x \, dx \). This step allows us to express the entire integral in terms of \( u \) and \( du \).
- Replace the original variables in the integral with these new terms. The integral \( \int e^x \sqrt{1+e^x} \, dx \) transitions to \( \int \sqrt{u} \, du \). This substitution reduces complexity, leaving us with a simpler integrand.
Power Rule Integration
Power rule integration is one of the fundamental rules in integral calculus. It helps us integrate functions of the form \( u^n \), where \( n \) is any real number. This rule simplifies integration, especially after substitution clarifies the integral鈥檚 form.
Here's the power rule in action:
Here's the power rule in action:
- For any \( n eq -1 \), the integral \( \int u^n \, du \) is given by \( \frac{u^{n+1}}{n+1} \). This rule is derived from reversing the power rule of differentiation.
- In our integral \( \int \sqrt{u} \, du \), we recognize \( \sqrt{u} \) as \( u^{1/2} \). Applying the power rule, \( \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} \), which simplifies to \( \frac{2}{3}u^{3/2} \).
- Remember, this power rule doesn't apply when \( n = -1 \). In such a case, the integral results in a natural logarithm function, \( \ln|u| \), instead.
Constant of Integration
The constant of integration, often represented by \( C \), is an essential part of indefinite integrals. It acts as a reminder of the many possible antiderivatives that could exist.
Here's why the constant of integration is important:
Here's why the constant of integration is important:
- Indefinite integrals represent a family of functions. The antiderivative isn't unique, as adding a constant to a function results in the same derivative. Therefore, \( C \) accounts for all these possibilities.
- In our evaluated integral \( \frac{2}{3} (1 + e^x)^{3/2} \), adding \( C \) signifies the complete solution \( \frac{2}{3} (1 + e^x)^{3/2} + C \).
- This constant becomes particularly significant when solving differential equations or initial value problems, as it can be determined by the initial conditions provided.
