Chapter 11: Problem 889
\(\int[\mathrm{dx} /(1+\tan \mathrm{x})]=\underline{ }+\mathrm{c}\) (a) \(\log |\sec x+\tan x|\) (b) \(2 \sec ^{2}(\mathrm{x} / 2)\) (c) \(\log \mid \mathrm{x}+\sin \mathrm{x}\) (d) \((1 / 2)[x+\log |\sin x+\cos x|]\)
Short Answer
Expert verified
None of the given options is correct.
Step by step solution
01
Set up the integral
Start by writing out the integral. In this case, the function to be integrated is \( \frac{dx}{1+\tan x} \).
02
Try a substitution
Set \( u = \tan x \), which implies \( du = \sec^2 x \, dx \). Substitute these values into the integral, giving \( \int \frac{1}{1 + u} \, du \).
03
Recognize a standard integral
This integral is a standard one – it is the integral of \( \frac{1}{1 + u} \), which is \( \ln |1+u| \).
04
Replace \(u\) with original value
Now that the integral has been solved, replace \( u \) with the original expression in terms of \( x \), i.e., \( \tan x \). So the antiderivative of \( \frac{1}{1 + \tan x} \) is \( \ln|1 + \tan x| \).
05
Return to Options
Compare this with the options provided.
(a) \(\log |\sec x+\tan x|\)
(b) \(2 \sec ^{2}(\mathrm{x} / 2)\)
(c) \(\log \mid \mathrm{x}+\sin \mathrm{x}\)
(d) \((1 / 2)[x+\log |\sin x+\cos x|]\)
From these options, none of them matches the solution, \( \ln|1 + \tan x| \). This means that none of the options (a,b,c,d) is the correct answer to the given integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration is a fundamental concept in calculus that allows us to find areas under curves, among other applications. There are several integration techniques that can be employed depending on the function you are working with. These methods help in transforming a difficult integral into one that is easier to evaluate.
- Substitution Method: This involves changing the variable of integration to simplify the integral. It is particularly useful when the integral includes a composite function. For example, in the problem provided, the substitution of variables from function of \(x\) into \(u\) simplifies the integration process.
- Integration by Parts: This is used primarily when the integrand is a product of two functions. The formula used here originates from the product rule for differentiation. Although not used in the given problem, it is an essential technique in integration.
- Partial Fractions: This is used for integrals involving rational functions, which can be expressed as the sum of simpler fractions. Again, not directly applicable to the given integral but fundamentally important.
Trigonometric Substitution
Trigonometric substitution is a powerful technique used when the integrand involves expressions like \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), or \(\sqrt{x^2 - a^2}\). This technique leverages trigonometric identities to simplify such expressions into a form that's easier to integrate.In the solved example, a slightly different yet related approach was used where \(u = \tan x\). By substituting \(\tan x\) and dealing with the trigonometric identity \(du = \sec^2 x \, dx\), it allowed simplification:
- The expression \(1 + \tan x\) was transformed into a polynomial form that could be integrated directly.
- Recognizing \(1 + \tan^2 x = \sec^2 x\) was key in simplifying the derivative during substitution, although not directly visible in the expression. This step is crucial for converting a more complex trigonometric component into an algebraic form that is easier to integrate.
Standard Integrals
Standard integrals refer to commonly seen integral forms that we often memorize for convenience, reducing the need to work them out each time from scratch. Recognizing such patterns helps in swiftly evaluating integrals.In our problem, after substituting \(u = \tan x\), the integral transformed into \(\int \frac{du}{1+u}\). This is a standard integral form whose antiderivative is known:
- \( \ln |1+u| \)
- Standard integrals allow fast and efficient solutions.
- They provide a reliable method to check work when tackling complex integrals.
- Having these expressions at your fingertips aids in focusing on more complex parts of the integration process.