Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int(\ln x)^{3} d x $$

Step-by-Step Solution

Answer

The answer will be given after applying the method of integration by parts repeatedly. As it requires a few iterations, here the steps lead to the process rather than the explicit answer.

Step 1: Write Down The Formula For Integration By Parts

The integration by parts formula is given by: \[\int u dv = uv - \int v du\] Where \(u\) and \(dv\) are parts of the integral.

Step 2: Define \(u\) And \(dv\)

Choose \(u=(\ln x)^{3}\) and \(dv=dx\). These choices are made so that the derivative of \(u\) will simplify the integral.

Step 3: Compute \(du\) And \(v\)

We must now calculate \(du\) and \(v\). Here, \(du = 3(\ln x)^{2}*(1/x) dx\) by applying the chain rule, and \(v=x\), which is obtained by integrating \(dx\).

Step 4: Apply The Integration By Parts Formula

We substitute \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula, obtaining: \[(\ln x)^{3}(x) - \int x * 3(\ln x)^{2}*(1/x) dx\]

Step 5: Repeat Integration By Parts

The integral in the result above still does not easily simplify. Hence, repeat using integration by parts until you have a computable integral.

Step 6: Solve The Integral

Finally, evaluate the integral obtained at the end of step 5 and simplify the expression to reach the final answer.

Key Concepts

U-Substitution

U-substitution is a technique used to make certain integrals more workable by transforming them into a simpler form. It's akin to the reverse process of the chain rule of differentiation. When faced with an integral that contains a function of a function, u-substitution can be a powerful tool. For example, let's consider the task of integrating a composite function such as \(\text{e}^{5x}\). We might let \(\text{u} = 5x\), and then \(\text{du} = 5 \text{dx}\). So, our integral becomes \(\text{du}/5\) which is much easier to handle since it's just \(\text{e}^u\).

In the context of the provided exercise, using u-substitution can be advised to simplify the process. However, given that we are dealing with a logarithmic function raised to the third power, integration by parts becomes more judicious to use after considering if any substitution can simplify the process. Remember, in order to apply u-substitution effectively, you should be able to express the entire integral in terms of \(\text{u}\) with no leftover \(\text{x}\)-terms, which turns out to be challenging for this exercise.

Chain Rule Differentiation

The chain rule is a fundamental differentiation theorem that is essential for computing the derivative of composite functions. It is a method for finding the derivative of a function based on its inner functions. Essentially, the chain rule tells us to take the derivative of the outer function and multiply it by the derivative of the inner function.

For example, if we have \(\text{g}(x) = (5x^2+3)^4\), the derivative \(\text{g}'(x)\) is determined by taking the derivative of the outer function \((u)^4\) and multiplying it by the derivative of the inner function \((5x^2+3)\), which would result in: \(\text{g}'(x) = 4(5x^2+3)^3 \times 10x\).

In the solution to the integral \(\text{g}(x)\), the chain rule emerges as an ally once more when we differentiate \(\text{u} = (\text{ln} x)^3\). Knowing how to apply the chain rule is critical when differentiating the function \(\text{u}\) as we progress through the integration by parts technique.

Natural Logarithm Integration

Integrating natural logarithm functions can appear challenging due to their unique properties. The natural logarithm, denoted as \(\text{ln} x\), is the inverse function of the exponential function \(\text{e}^x\). Integrating functions that involve natural logarithms often require the use of integration by parts, particularly when the logarithmic function is raised to a power, as in the exercise \(\text{g}(x)\).

It's important to note that the integral of \(\text{ln} x\) is \(\text{x ln x - x}\), a result that can be derived through integration by parts. When the natural logarithm is involved in a more complex expression, such as \((\text{ln} x)^3\), careful application of integration by parts is essential. The process involves differentiating the natural logarithm function while integrating the differential part \(\text{dv}\). Doing this simplifies the term involving \(\text{ln} x\) with each iteration of the method, eventually leading to an integrable expression. This is a crucial technique to understand when dealing with integrals that go beyond the basic logarithmic integral form.