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Chapter 28: Problem 24

Integrate each of the given functions. $$\int \frac{e^{x} d x}{e^{2 x}+3 e^{x}+2}$$

Short Answer

Expert verified
The integral simplifies to a combination of logarithmic functions after partial fraction decomposition and integration.

Step by step solution

01

Identify the Structure

Notice that the denominator can be factored. The expression in the denominator is a quadratic in terms of \(e^{x}\).
02

Substitute Variable

Let \( u = e^x \). Then \( du = e^x \, dx \). This implies that \( dx = \frac{du}{u} \). Substitute into the integral: \[ \int \frac{du}{u(u^2 + 3u + 2)} \]
03

Factor the Denominator

The quadratic \(u^2 + 3u + 2\) factors into \((u+1)(u+2)\). Thus, the integral becomes: \[ \int \frac{du}{u(u+1)(u+2)} \]
04

Apply Partial Fraction Decomposition

Express \(\frac{1}{u(u+1)(u+2)}\) as a sum of partial fractions: \[ \frac{1}{u(u+1)(u+2)} = \frac{A}{u} + \frac{B}{u+1} + \frac{C}{u+2} \]Multiply through by the common denominator to solve for \(A\), \(B\), and \(C\).
05

Solve for Coefficients

Solve the resultant equation from the previous step: Set up system of equations:\[A(u+1)(u+2) + B(u)(u+2) + C(u)(u+1) = 1\]By expanding and equating coefficients, solve for \(A, B, C\).
06

Integrate Each Term

After solving for \( A, B, \) and \( C \), integrate each term separately:\[A\int \frac{1}{u} \, du + B\int \frac{1}{u+1} \, du + C\int \frac{1}{u+2} \, du\]Each of these integrals is a standard form that results in logarithmic functions.
07

Back-Substitute the Original Variable

Once integration is complete, replace \( u \) with \( e^x \) to revert back to the original variable: The integral becomes a combination of logarithmic expressions in terms of \( e^x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial fraction decomposition
Partial fraction decomposition is a powerful technique in integration that involves breaking down a complex fraction into a sum of simpler fractions. This simplifies the integration process. In the exercise, the integral contains a denominator that can be factored into simpler terms.

Here is how partial fraction decomposition works:
  • First, express the original complex fraction as a sum of simpler fractions.
  • Assign coefficients (such as \( A, B, \) and \( C \)) to these fractions.
  • Multiply through by the common denominator to get a polynomial equation.
  • Finally, solve the system of equations to find the values of these coefficients.
Once the coefficients are determined, you can integrate each term individually. This step makes the problem much more manageable, especially when towards integrating rational functions.
Exponential functions
Understanding exponential functions is crucial in this exercise as the function to integrate includes an exponential component, \( e^x \). The exponential function \( e^x \) is unique:
  • It is its own derivative, meaning the rate of change is proportional to the value of the function itself.
  • Its integral formula is straightforward: \( \int e^x \, dx = e^x + C \), where \( C \) is the constant of integration.
In the context of this particular problem, recognizing that \( e^{2x} \) can also be expressed using exponential properties \((e^x)^2\) is necessary. This allows for substitution and simplification of the integral. Exponential functions are key in calculus, representing continual growth or decay, and play essential roles in various scientific disciplines.
Substitution method
The substitution method is a widely used integration technique where you change variables to simplify the integration process. It's particularly helpful when dealing with complex integrals.

In this exercise, the substitution \( u = e^x \) is used:
  • This substitution transforms \( e^x \) into \( u \), leading to a simpler form for manipulation.
  • The derivative \( du = e^x \, dx \) provides the information needed to change \( dx \) into \( \frac{du}{u} \).
By doing so, the integral becomes a rational function of \( u \), which can then be easily tackled using partial fraction decomposition. Substitution simplifies complicated integrals by reformatting them into more easily solvable forms.

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Most popular questions from this chapter

identify the form of each integral as being inverse sine, inverse tangent, logarithmic, or general power, as in Examples 6 and 7. Do not integrate. In each part (a), explain how the choice was made. (a) \(\int \frac{2 d x}{9 x^{2}+4}\) (b) \(\int \frac{2 x d x}{\sqrt{9 x^{2}-4}}\) (c) \(\int \frac{2 x d x}{9 x^{2}-4}\)

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Solve the given problems by integration. Under specified conditions, the time \(t\) (in min) required to form \(x\) grams of a substance during a chemical reaction is given by \(t=\int d x /[(4-x)(2-x)] .\) Find the equation relating \(t\) and \(x\) if \(x=0\) g when \(t=0\) min.

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