Problem 21
Question
Find the indefinite integral and check the result by differentiation. $$ \int \frac{x-2}{\sqrt{x^{2}-4 x+3}} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral \(\int \frac{x-2}{\sqrt{x^{2}-4 x+3}} d x\) is \(\ln|(x - 2) + \sqrt{(x - 2)^2 - 1}|\)
1Step 1: Complete The Square
To simplify the quadratic inside the square root, complete the square in the denominator by rewriting \(x^2 - 4x + 3\) as \((x - 2)^2 - 1\) or \(x^2 - 4x + 4 - 1\)
2Step 2: Substitute
Now, let \(u = x - 2\), and then \(du = dx\), so we get the integral \(\int \frac{u}{\sqrt{u^{2} - 1}} du\)
3Step 3: Use Trigonometric Substitution
The trigonometric substitution \(u = \sec(t)\) simplifies the integral. The reason is that \(\sqrt{u^{2} - 1}\) simplifies to \(\tan(t)\) when \(u = \sec(t)\). The differential \(du = \sec(t)\tan(t)dt\)
4Step 4: Perform The Integration
Plug the substitutions \(u = \sec(t)\) and \(du = \sec(t)\tan(t)dt\) into the integral, which then simplifies to \(\int \sec(t) dt\). The antiderivative of \(\sec(t)\) is \(\ln|sec(t) + \tan(t)|\).
5Step 5: Back-substitution
Replace \(t\) with \(\sec^{-1}(u)\) to obtain \(\ln|\sec(\sec^{-1}(u)) + \tan(\sec^{-1}(u))|\). This simplifies to \(\ln|u + \sqrt{u^{2} - 1}|\). Then substitute \(x - 2\) for \(u\) to get the anti-derivative \(\ln|(x - 2) + \sqrt{(x - 2)^2 - 1}|\)
6Step 6: Checking the integral
Differentiate \(\ln|(x - 2) + \sqrt{(x - 2)^2 - 1}|\) and it should return the integrand \(\frac{x - 2}{\sqrt{x^{2} - 4x + 3}}\) as required for the problem
Key Concepts
Trigonometric SubstitutionIntegration TechniquesDifferentiation Check
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify the integration process, especially when dealing with square roots involving quadratic expressions. In this problem, we work with the integral \( \int \frac{x-2}{\sqrt{x^{2}-4 x+3}} \, dx \).
First, we need to complete the square in the expression inside the square root to transform \(x^2 - 4x + 3\) into \((x-2)^2 - 1\). This step makes it easier to identify the trigonometric identity that best fits the substitution. By setting \(u = x - 2\), the integral becomes \( \int \frac{u}{\sqrt{u^{2} - 1}} \, du \).
Here, we opt for a substitution that integrates well with the identity \( \sec^2(t) - 1 = \tan^2(t) \). Thus, setting \(u = \sec(t)\) transforms \(\sqrt{u^2 - 1}\) into \(\tan(t)\), which can simplify the integral considerably. This is because the derivative \(du = \sec(t) \tan(t) \, dt\) fits naturally, aligning our integral for easier evaluation.
First, we need to complete the square in the expression inside the square root to transform \(x^2 - 4x + 3\) into \((x-2)^2 - 1\). This step makes it easier to identify the trigonometric identity that best fits the substitution. By setting \(u = x - 2\), the integral becomes \( \int \frac{u}{\sqrt{u^{2} - 1}} \, du \).
Here, we opt for a substitution that integrates well with the identity \( \sec^2(t) - 1 = \tan^2(t) \). Thus, setting \(u = \sec(t)\) transforms \(\sqrt{u^2 - 1}\) into \(\tan(t)\), which can simplify the integral considerably. This is because the derivative \(du = \sec(t) \tan(t) \, dt\) fits naturally, aligning our integral for easier evaluation.
Integration Techniques
Integration involves various approaches and techniques to solve different types of problems. One prominent method seen here is the application of trigonometric substitution to simplify challenging integrals.
After substituting \(u = \sec(t)\), the integral transforms into \( \int \sec(t) \, dt \). Known methods of integration, such as recognizing standard integrals, come into play at this stage. The antiderivative of \(\sec(t)\) is a standard result: it is given by \( \ln |\sec(t) + \tan(t)| \), based on a commonly derived integration formula.
Using these techniques not only simplifies the integration process but also reinforces one's understanding of identity interplay and manipulation to solve integrals effectively. It's crucial to be familiar with these identities and practice them frequently, as they prove useful in solving a wide array of integral problems.
After substituting \(u = \sec(t)\), the integral transforms into \( \int \sec(t) \, dt \). Known methods of integration, such as recognizing standard integrals, come into play at this stage. The antiderivative of \(\sec(t)\) is a standard result: it is given by \( \ln |\sec(t) + \tan(t)| \), based on a commonly derived integration formula.
Using these techniques not only simplifies the integration process but also reinforces one's understanding of identity interplay and manipulation to solve integrals effectively. It's crucial to be familiar with these identities and practice them frequently, as they prove useful in solving a wide array of integral problems.
Differentiation Check
Finally, verifying the correctness of the indefinite integral solution is crucial. This is typically done using differentiation. Once we find the antiderivative, the next step is to differentiate it to ensure we return to the original integrand.
In our case, the derived antiderivative \( \ln|(x - 2) + \sqrt{(x - 2)^2 - 1}| \) needs to differentiate back to the integrand, \( \frac{x - 2}{\sqrt{x^{2} - 4x + 3}} \). By performing the differentiation here, we apply the chain rule correctly and simplify accordingly.
Hence, this differentiation check assures us of the accuracy of the entire integration process.
In our case, the derived antiderivative \( \ln|(x - 2) + \sqrt{(x - 2)^2 - 1}| \) needs to differentiate back to the integrand, \( \frac{x - 2}{\sqrt{x^{2} - 4x + 3}} \). By performing the differentiation here, we apply the chain rule correctly and simplify accordingly.
- Differentiate the outer function \(\ln|u + \sqrt{u^2 - 1}|\) to get \(\frac{1}{u + \sqrt{u^2 - 1}}\).
- Next, multiply this by the derivative of the inner function \(u + \sqrt{u^2 - 1}\).
- It confirms that it simplifies back to the original integrand.
Hence, this differentiation check assures us of the accuracy of the entire integration process.
Other exercises in this chapter
Problem 21
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered}
View solution Problem 21
Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{x^{3}+1} d x $$
View solution Problem 22
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{e^{1 / x}}{x^{2}}, y=0, x=1, x=3 $$
View solution Problem 22
Use the Log Rule to find the indefinite integral. $$ \frac{x}{x^{2}+4} d x $$
View solution