Problem 52
Question
Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{4} \ln x d x \quad \text { Integration by parts } $$
Step-by-Step Solution
Verified Answer
The integral of \(x^{4}\ln x dx\) is \((1/5)x^{5}\ln x - (1/25)x^{5} + C\), where C is the constant of integration.
1Step 1: Set up the Integral Using the Formula for Integration by Parts
In the formula for integration by parts: \(\int u dv = u v - \int v du\), if we let \(u = \ln x\) and \(dv = x^{4} dx\), we can derive \(du = (1/x) dx\) and \(v = x^{5}/5\).
2Step 2: Substitute u, v, du, and dv into the Integration by Parts Formula
The integration by parts formula \(uv - \int v du\) becomes \(\ln x * x^{5}/5- \int (x^{5}/5 * (1/x) dx)\) which simplifies to \(\ln x * x^{5}/5- \int x^{4}/5 dx\).
3Step 3: Evaluate the Remaining Integral
The integral in the formula is a basic power rule integral. The integral \(∫x^{4} dx = x^{5}/5\), so the equation becomes \(\ln x * x^{5}/5 - x^{5}/25 = (1/5)x^{5}\ln x - (1/25)x^{5}\).
Key Concepts
Indefinite IntegralIntegration TechniquesPower Rule for Integration
Indefinite Integral
An indefinite integral represents a family of functions whose derivative is the original function being integrated. In more practical terms, it is the antiderivative of a given function. Performing an indefinite integral on a function gives us a result that includes all the possible functions which, when differentiated, would yield the original function, plus an arbitrary constant 'C' to account for all possible vertical shifts. The symbol used to denote an indefinite integral is the integral sign without upper or lower limits.
For instance, if you are tasked with finding the indefinite integral of a function like \( f(x) = x^2 \), you would seek a function whose derivative is \( x^2 \). In this case, the indefinite integral, often called the antiderivative, would be \( F(x) = \frac{x^3}{3} + C \), where 'C' represents the constant of integration.
For instance, if you are tasked with finding the indefinite integral of a function like \( f(x) = x^2 \), you would seek a function whose derivative is \( x^2 \). In this case, the indefinite integral, often called the antiderivative, would be \( F(x) = \frac{x^3}{3} + C \), where 'C' represents the constant of integration.
Integration Techniques
Integration techniques are various methods used to find the integral of a function that may not be straightforward to integrate. One common method is the 'power rule'. For more complex functions, methods such as 'integration by parts,' 'substitution,' 'partial fractions,' or 'trigonometric integrals' might be employed depending on the form of the function to be integrated.
For example, with the problem \( \int x^4 \ln x \, dx \), we recognize that the presence of both a polynomial term and a logarithmic function suggests using integration by parts, a technique that is particularly useful when integrating the product of two functions that are not easily integrated on their own. It allows us to break down the integral into simpler parts that we can handle individually.
For example, with the problem \( \int x^4 \ln x \, dx \), we recognize that the presence of both a polynomial term and a logarithmic function suggests using integration by parts, a technique that is particularly useful when integrating the product of two functions that are not easily integrated on their own. It allows us to break down the integral into simpler parts that we can handle individually.
Power Rule for Integration
The power rule for integration is one of the most basic, yet vital, integration techniques. It states that the integral of \( x^n \), where \( n \) is any real number except -1, is given by \( \frac{x^{n+1}}{n+1} \) plus the constant of integration.
To apply this rule, increase the exponent of \( x \) by 1 and divide by this new exponent. However, remember that this rule does not apply when \( n = -1 \) because the integral of \( 1/x \) is \( \ln|x| \) instead. In our exercise, when we simplify the integration by parts result, we get a power function \( x^4 \) that we need to integrate. This is where the power rule neatly comes into play, allowing us to integrate this remaining term to get \( x^5/5 \), thus moving us towards the final solution to the problem.
To apply this rule, increase the exponent of \( x \) by 1 and divide by this new exponent. However, remember that this rule does not apply when \( n = -1 \) because the integral of \( 1/x \) is \( \ln|x| \) instead. In our exercise, when we simplify the integration by parts result, we get a power function \( x^4 \) that we need to integrate. This is where the power rule neatly comes into play, allowing us to integrate this remaining term to get \( x^5/5 \), thus moving us towards the final solution to the problem.
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Problem 52
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