Problem 19
Question
Use substitution to evaluate the indefinite integrals. $$ \int(4 x-3) \sqrt{2 x^{2}-3 x+2} d x $$
Step-by-Step Solution
Verified Answer
\(\frac{2}{3} (2x^2 - 3x + 2)^{3/2} + C\)
1Step 1: Identify Substitution Variable
The expression under the square root, \(2x^2 - 3x + 2\), is complex, making it a candidate for substitution. Let's set \(u = 2x^2 - 3x + 2\).
2Step 2: Differentiate u with respect to x
Find \(\frac{du}{dx}\) to help substitute the differential. We differentiate: \(\frac{du}{dx} = 4x - 3\). Thus, \(du = (4x - 3) dx\).
3Step 3: Perform the Substitution
Substitute \(u = 2x^2 - 3x + 2\) and \(du = (4x - 3) dx\) into the integral. This replaces the integral entirely, yielding: \[\int \sqrt{u} \, du\].
4Step 4: Integrate with respect to u
The integral \(\int \sqrt{u} \, du\) is now simpler. Rewrite \(\sqrt{u}\) as \(u^{1/2}\), and integrate: \[\int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C\].
5Step 5: Substitute Back the Original Expression
Now substitute back \(u = 2x^2 - 3x + 2\) into the integrated function: \[\frac{2}{3} (2x^2 - 3x + 2)^{3/2} + C\].
Key Concepts
Understanding Indefinite IntegralsBasics of DifferentiationEssential Integration Techniques
Understanding Indefinite Integrals
Indefinite integrals are fundamentally about finding functions from their derivatives. They do not provide a specific value, but rather a family of functions, represented generally by adding the constant 'C' at the end of integration. This constant signifies that there are multiple functions that can represent an indefinite integral, each differing by some constant value.
When you see an integral sign without upper and lower bounds, like \( \int f(x) \, dx \), you are dealing with an indefinite integral. The goal here is to "reverse" differentiation—transforming the rate of change back into the original function. This process is often called anti-differentiation.
Indefinite integrals are crucial because they help us understand how a function accumulates values. They tell us the underlying function given its rate of change. For example, in physics, if you're given acceleration, which is the rate of change of velocity, indefinite integration can recover the original velocity data.
When you see an integral sign without upper and lower bounds, like \( \int f(x) \, dx \), you are dealing with an indefinite integral. The goal here is to "reverse" differentiation—transforming the rate of change back into the original function. This process is often called anti-differentiation.
Indefinite integrals are crucial because they help us understand how a function accumulates values. They tell us the underlying function given its rate of change. For example, in physics, if you're given acceleration, which is the rate of change of velocity, indefinite integration can recover the original velocity data.
Basics of Differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents how a function changes over a very small interval. Simply put, it measures the rate at which one variable changes with another. For a function \(y = f(x)\), its derivative \(f'(x)\) indicates how \(y\) changes as \(x\) changes.
The process involves several rules such as the power rule, product rule, quotient rule, and chain rule, each helping to simplify the finding of derivatives in different function scenarios. In our exercise, after choosing a substitution, we use differentiation to transform the differential part of the integral, enabling us to make the substitution simpler and easier to integrate.
In this exercise, differentiating \(u = 2x^2 - 3x + 2\) with respect to \(x\) gave us \(du = (4x - 3) dx\). This illustrates the power of differentiation in helping to convert complex expressions into more manageable forms. By doing so, you prepare the groundwork for smoother integration.
The process involves several rules such as the power rule, product rule, quotient rule, and chain rule, each helping to simplify the finding of derivatives in different function scenarios. In our exercise, after choosing a substitution, we use differentiation to transform the differential part of the integral, enabling us to make the substitution simpler and easier to integrate.
In this exercise, differentiating \(u = 2x^2 - 3x + 2\) with respect to \(x\) gave us \(du = (4x - 3) dx\). This illustrates the power of differentiation in helping to convert complex expressions into more manageable forms. By doing so, you prepare the groundwork for smoother integration.
Essential Integration Techniques
Integration techniques are strategies or methods developed to solve integrals more efficiently. Often, the initial form of an integral is too complex, so you need to apply these techniques to make integration manageable. A common and powerful technique is substitution, which is showcased in the exercise here.
Substitution is like reverse chain rule; it involves changing variables to simplify the integral. The basic idea is to identify an inner function within the integral and substitute it with a single variable, typically \(u\).
Substitution is like reverse chain rule; it involves changing variables to simplify the integral. The basic idea is to identify an inner function within the integral and substitute it with a single variable, typically \(u\).
- Choose a substitution: Identify part of the integral that could be swapped for a variable 'u', simplifying the expression.
- Differentiate u: To replace \(dx\), differentiate the 'u' substitution with respect to 'x'.
- Perform the substitution: Replace both the function and differential parts of the integral with 'u' and 'du'.
- Integrate with respect to 'u': This often results in a much simpler integral.
- Substitute back: Once integrated, replace 'u' with the original expression in terms of 'x'.
Other exercises in this chapter
Problem 18
In Problems , use partial-fraction decomposition to evaluate the integrals.$$ \int \frac{4 x^{2}-x-1}{(x+1)^{2}(x-3)} d x $$
View solution Problem 19
Use integration by parts to evaluate the integrals. $$ \int_{1}^{2} \ln x d x $$
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In Problems 19-23, compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given
View solution Problem 19
Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{4}} d x $$
View solution