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Magazine Chapter 9: Problem 15
Evaluate the integral. $$ \int \frac{e^{2 x}}{e^{x}+4} d x $$
Short Answer
Expert verified
\( \frac{e^{2x}}{2} - e^x - 16 + 16 \ln{|e^x + 4|} + C \)
Step by step solution
01
Select a Substitution
To evaluate the given integral \( \int \frac{e^{2x}}{e^x + 4} \, dx \), we will use substitution. Let \( u = e^x + 4 \). Consequently, the derivative \( du = e^x \, dx \). This substitution will help simplify the integral.
02
Express \( e^{2x} \, dx \) in Terms of \( u \) and \( du \)
To substitute \( e^{2x} \, dx \) in terms of \( u \), express it using \( e^x \): \( e^{2x} = (e^x)^2 = (u - 4)^2 \). Thus, \( e^{2x} \, dx = (u - 4)^2 \, du \). Now, substitute this and the expression for \( dx \) into the integral.
03
Rewrite the Integral
Substituting into the integral, we have: \[ \int \frac{(u - 4)^2}{u} \, du. \] Simplify this expression by dividing:\[ \int \left( u - 8 + \frac{16}{u} \right) \, du. \]
04
Evaluate the Simplified Integral
Integrate term by term:- For \( \int u \, du \), use \( \frac{u^2}{2} \).- For \( \int 8 \, du \), use \( 8u \).- For \( \int \frac{16}{u} \, du \), use \( 16 \ln{|u|} \).Combine these results into:\[ \frac{u^2}{2} - 8u + 16 \ln{|u|} + C. \]
05
Back-Substitute \( u = e^x + 4 \)
Replace \( u \) with \( e^x + 4 \) to translate back into terms of \( x \):\[ \frac{(e^x + 4)^2}{2} - 8(e^x + 4) + 16 \ln{|e^x + 4|} + C. \]
06
Simplify the Expression
Expand \((e^x + 4)^2\) to \(e^{2x} + 8e^x + 16\) and substitute:\[ \frac{1}{2}(e^{2x} + 8e^x + 16) - 8e^x - 32 + 16 \ln{|e^x + 4|} + C. \]This simplifies to:\[ \frac{e^{2x}}{2} - e^x - 16 + 16 \ln{|e^x + 4|} + C. \]
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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 powerful technique used in integral calculus to simplify complex integrals. It involves replacing a part of the integral with a new variable, called the substitution variable, which transforms the integral into an easier form.
- Firstly, identify a substitution that will simplify the integrand. For example, if you encounter an expression like \( e^x + 4 \), setting it equal to a new variable \( u \) can be beneficial.
- Next, compute the derivative of your substitution: if \( u = e^x + 4 \), then \( du = e^x \, dx \).
- Replace the original expression in the integral with terms of \( u \): this might mean rewriting \( e^{2x} \, dx \) as \((u - 4)^2 \, du \) if needed.
Integral Calculus
Integral calculus is one of the two main branches of calculus. It is primarily concerned with the concept of integrals, which represent the area under a curve or the accumulation of quantities.
- Integrals can be definite, where they have specific limits of integration, or indefinite, where they lack bounds.
- In solving integrals, algebraic simplification and applying methods like substitution are crucial to finding solutions.
- Integration techniques often involve breaking down complex expressions, simplifying them, and computing the antiderivatives.
Exponential Functions
Exponential functions, especially involving the natural base \( e \), play a significant role in calculus. Exponential functions are of the form \( e^x \), where the rate of growth is proportional to the value itself.
- In integral calculus, these functions often appear as part of the functions to be integrated, due to their properties and frequent occurrence in natural phenomena.
- The exponential function \( e^x \) and its powers, such as \( e^{2x} \), are particularly common as they simplify or reshape into expressions that can be more easily handled, especially when combined with substitution.
- These functions maintain their core identity even after differentiation or integration, such as \( \int e^x \, dx = e^x + C \).
