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Chapter 7: Problem 70

(a) Make \(u\) -substitution ( 5) to convert the integrand to a rational function of \(u,\) and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$\int \frac{\sin x}{\sin x+\tan x} d x$$

Short Answer

Expert verified
Using substitution is complex; a CAS is better for direct evaluation.

Step by step solution

01

Identify the substitution

First, decide on a substitution that will simplify the integrand. Given the expression is \( \frac{\sin x}{\sin x + \tan x} \), we can use \( u = \sin x + \tan x \). This choice is intended to simplify the denominator.
02

Differentiate the substitution

Differentiate the chosen substitution with respect to \( x \). Starting with \( u = \sin x + \tan x \), the derivative is \( du = (\cos x + \sec^2 x) dx \). This will be useful to change the variables in the integral.
03

Express \( dx \) in terms of \( du \)

Rearrange \( du = (\cos x + \sec^2 x) dx \) to express \( dx \) in terms of \( du \). This gives \( dx = \frac{du}{\cos x + \sec^2 x} \).
04

Replace \( x \) in the integral with \( u \) terms

Substitute \( \sin x \) and \( \tan x \) with expressions involving \( u \) in the integral. However, this process becomes complex, and alternative approaches may be necessary. Proceed to calculate, recognizing the difficulty involved and needing further insight to simplify.
05

Evaluate the integral using a CAS

Since manual substitution is complex, use a Computer Algebra System (CAS) to evaluate the integral directly. A CAS can provide a more straightforward solution without manual substitutions.
06

Verify and compare results

Compare the result obtained from the substitution method (if solved through simplification) with the CAS result. This confirms the accuracy of both methods.

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Key Concepts

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

U-substitution
U-substitution is a common technique in calculus used to simplify the process of integration, especially when dealing with complex functions. Essentially, it involves changing the variable of integration to transform the integrand to a more manageable form.
  • Start by selecting a part of the integrand to substitute as a new variable, often denoted by \( u \).
  • Choose \( u \) such that differentiating \( u \) will simplify the integration process.
  • Compute the derivative \( du \) in terms of \( dx \).
The goal is to rewrite the integral in terms of \( u \) rather than \( x \), facilitating easier integration. In our original problem, choosing \( u = \sin x + \tan x \) was intended to simplify the denominator, though this might not always yield the simplest solution right away. Sometimes, further manipulation or different execution of the plan might be necessary to resolve the integral efficiently.
Integration by Substitution
Integration by substitution is essentially performing a change of variables. This method is analogous to the chain rule for differentiation and is one of the most commonly used techniques for integrals. Here鈥檚 how it is typically done:
  • Choose a substitution that simplifies the original integrand significantly.
  • Replace the entire integrand and \( dx \) with expressions involving the new variable \( u \).
  • Integrate the transformed expression with respect to \( u \).
  • Transform back to the original variable by substituting for \( u \).
In the exercise, identifying \( u = \sin x + \tan x \) led to the derivative used to express \( dx \) in terms of \( du \). Despite complexities in this particular problem, substitution helps focus the integrand into a form that can be more readily solved, often requiring subsequent algebraic manipulation.
Rational Functions
Rational functions are expressions that can be written as the ratio of two polynomials. These functions are often integrated using substitution to transform them into simpler rational expressions.
  • They typically appear in forms such as \( \frac{P(x)}{Q(x)} \), where both \( P \) and \( Q \) are polynomials.
  • Integration of rational functions might involve factoring the denominator or completing the square.
In our exercise, converting the given expression into a rational function of \( u \) was the objective. The substitution was chosen with the hope that the resulting function would become a straightforward rational expression, easy to integrate. In practice, choosing the appropriate substitution is key and will guide how the rational function behaves under the substitution.
Computer Algebra Systems (CAS)
Computer Algebra Systems (CAS) are powerful tools that handle complex algebraic calculations, including integration. These systems can efficiently perform operations that might be difficult or time-consuming by hand.
  • They can solve integrals without manual substitution steps, providing exact or approximate solutions.
  • CAS can verify results by comparing different methods of integration.
In the given exercise, a CAS was used to verify the result obtained from manual substitution. This is particularly useful because it quickly confirms whether the manual process, potentially complicated by algebraic manipulation, is correct. By automatically evaluating complex integrals, CAS provides students and mathematicians alike with an essential tool for solving otherwise cumbersome problems.

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