Chapter 1: Problem 45
In the following exercises, integrate using the indicated substitution. $$ \int e^{2 x} \sqrt{1-e^{2 x}} d x ; u=e^{2 x} $$
Short Answer
Expert verified
The integral evaluates to \( \frac{1}{3} (1-e^{2x})^{3/2} + C \).
Step by step solution
01
Substitution and Change of Variables
We start by substituting the given variable. Let \( u = e^{2x} \). Differentiate both sides with respect to \( x \) to express \( dx \) in terms of \( du \). The derivative is \( du = 2e^{2x} dx \). Therefore, \( dx = \frac{du}{2u} \).
02
Express the Integral in Terms of u
Substitute \( u = e^{2x} \) and \( dx = \frac{du}{2u} \) into the integral. The integral becomes:\[ \int e^{2x} \sqrt{1-u} \frac{du}{2u} \]The term \( e^{2x} \) can be replaced by \( u \), giving:\[ \int \frac{u \sqrt{1-u} }{2u} \, du = \frac{1}{2} \int \sqrt{1-u} \, du \]
03
Integration with Power Formula
Use the power rule to integrate \( \sqrt{1-u} \) which can be rewritten as \( (1-u)^{1/2} \). Using the power formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), integrate:\[ \frac{1}{2} \int (1-u)^{1/2} \, du = \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} + C \]
04
Simplifying the Result
Simplify the result from the integration:\[ \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} = \frac{1}{3} (1-u)^{3/2} + C \]
05
Back-substitute Original Variable
Replace \( u \) back with \( e^{2x} \), as originally \( u = e^{2x} \):\[ \frac{1}{3} (1-e^{2x})^{3/2} + C \] This represents the complete solution to the initial integral.
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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 powerful technique used in integration when dealing with complex expressions. It involves substituting a part of the integrand with a new variable to simplify the integration process. In this context, we chose the substitution \( u = e^{2x} \) to replace the exponential component of the integrand.
This transformation helps break down the original function into forms that are easier to integrate. To apply the substitution correctly, we also need to express \( dx \) in terms of \( du \). By differentiating \( u = e^{2x} \), we find that \( du = 2e^{2x} \, dx \). Solving for \( dx \), we get \( dx = \frac{du}{2u} \).
This step is crucial because it allows us to rewrite the integral in terms of the new variable \( u \), thus making the integration process more straightforward.
This transformation helps break down the original function into forms that are easier to integrate. To apply the substitution correctly, we also need to express \( dx \) in terms of \( du \). By differentiating \( u = e^{2x} \), we find that \( du = 2e^{2x} \, dx \). Solving for \( dx \), we get \( dx = \frac{du}{2u} \).
This step is crucial because it allows us to rewrite the integral in terms of the new variable \( u \), thus making the integration process more straightforward.
Power Rule for Integration
The power rule for integration is a fundamental principle that assists in integrating functions of the form \( x^n \). The rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the integration constant. This is a direct application of the inverse process of differentiation.
When we applied the substitution \( u = e^{2x} \), our integral transformed into an expression involving \( \sqrt{1-u} \), which can be rewritten as \( (1-u)^{1/2} \).
Using the power rule, we integrated it as follows: \( \frac{1}{2} \int (1-u)^{1/2} \, du = \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} \).
This step is critical as it simplifies the integration into a manageable algebraic expression. The result underscores the significant role of the power rule in evaluating complex integrals derived from integration by substitution.
When we applied the substitution \( u = e^{2x} \), our integral transformed into an expression involving \( \sqrt{1-u} \), which can be rewritten as \( (1-u)^{1/2} \).
Using the power rule, we integrated it as follows: \( \frac{1}{2} \int (1-u)^{1/2} \, du = \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} \).
This step is critical as it simplifies the integration into a manageable algebraic expression. The result underscores the significant role of the power rule in evaluating complex integrals derived from integration by substitution.
Change of Variables
The change of variables is an essential step in the substitution method. It essentially transforms a difficult integral into a simpler one by modifying both the variable of integration and the integrand itself.
In our specific problem, we initially let \( u = e^{2x} \), which directly impacted the integrand, turning \( \int e^{2x} \sqrt{1-e^{2x}} \, dx \) into \( \frac{1}{2} \int \sqrt{1-u} \, du \). This transformation simplifies the problem because \( \sqrt{1-u} \) is more straightforward to integrate than the original expression containing \( e^{2x} \).
By effectively changing the variables, we facilitate the integration step processes and adjust our focus to more easily manageable algebraic forms. This strategic alteration is pivotal in completing the integration process efficiently and neatly.
In our specific problem, we initially let \( u = e^{2x} \), which directly impacted the integrand, turning \( \int e^{2x} \sqrt{1-e^{2x}} \, dx \) into \( \frac{1}{2} \int \sqrt{1-u} \, du \). This transformation simplifies the problem because \( \sqrt{1-u} \) is more straightforward to integrate than the original expression containing \( e^{2x} \).
By effectively changing the variables, we facilitate the integration step processes and adjust our focus to more easily manageable algebraic forms. This strategic alteration is pivotal in completing the integration process efficiently and neatly.
Expression Simplification
Expression simplification is all about making the equation or integral easier to handle. It involves rewriting it in a form that is more recognizable and solvable. In the context of this integration problem, simplifying the result after applying the substitution and integration steps is a critical phase.
Once we applied the integration via the power rule, we arrived at \( \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} \). Simplifying such expressions involves multiplying and rearranging parts to achieve a cleaner result. Here, the simplification yielded \( \frac{1}{3} (1-u)^{3/2} + C \).
This step is essential not only for the neatness of the solution but also for correctly interpreting the problem's answer. In the final stages, don't forget to substitute back \( u = e^{2x} \), ensuring the solution is in terms of the original variable. The final simplified form, \( \frac{1}{3} (1-e^{2x})^{3/2} + C \), represents the complete and tidy result, ready for interpretation or further application.
Once we applied the integration via the power rule, we arrived at \( \frac{1}{2} \cdot \frac{(1-u)^{3/2}}{3/2} \). Simplifying such expressions involves multiplying and rearranging parts to achieve a cleaner result. Here, the simplification yielded \( \frac{1}{3} (1-u)^{3/2} + C \).
This step is essential not only for the neatness of the solution but also for correctly interpreting the problem's answer. In the final stages, don't forget to substitute back \( u = e^{2x} \), ensuring the solution is in terms of the original variable. The final simplified form, \( \frac{1}{3} (1-e^{2x})^{3/2} + C \), represents the complete and tidy result, ready for interpretation or further application.
