Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{1}{x(\ln x)^{3}} d x $$

Short answer

The indefinite integral of \(\frac{1}{x(\ln x)^{3}} dx\) is \(-\frac{1}{2(\ln x)^{2}} + C\).

Step-by-step solution

Identify the Substitution

We can simplify the integral by substituting \(u = \ln x\). This reduces the complexity of the function inside the integral.

Differentiate the Substitution

Once you have identified the function to substitute, find its derivative. The derivative of \(u = \ln x\) is \(du = \frac{1}{x} dx\). This is useful as it will replace the \(\frac{1}{x} dx\) part in the original integral.

Substitute into the Integral

Now, substitute the \(u\) and its derivative into the integral. So, this gives us \(\int \frac{1}{u^{3}} du\). This is a simpler integral to solve.

Integrate

Integrate the function above with respect to \(u\). This integration is straight forward and gives \(-\frac{1}{2u^{2}} + C\), where \(C\) represents the constant of integration.

Reverse Substitution

Substitute back \(u = \ln x\) into the equation to get the final solution. This gives \(-\frac{1}{2(\ln x)^{2}} + C\)

Key concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify integration problems. It involves changing the variable of integration from one that might be complex to a simpler one. In essence, you're making the problem easier by temporarily substituting a more complicated part of the integral with a simpler variable.

To use substitution effectively, begin by identifying a part of the integrand that can be replaced with a new variable. This often involves a composite function or a logarithmic expression, as seen in cases involving the natural logarithm function, such as \(\ln x\). Here are some key steps:

• *Choose a substitution:* Identify which part of the expression would simplify the integral when substituted by a new variable (e.g., set \(u = \ln x\)).
• *Differentiate the substitution:* Take the derivative of your chosen \(u\) to help express the rest of the integral in terms of \(du\) (e.g., \(du = \frac{1}{x} dx\)).

By using the substitution method, you change the integral into a format that is often much more manageable, making it easier to solve and integrate with respect to the new variable.

Logarithmic Functions

Logarithmic functions, particularly the natural logarithm \(\ln x\), play a vital role in calculus and integration. The natural logarithm is the inverse of the exponential function, which inherently links various mathematical concepts.

When working with logarithmic functions in integration, you often encounter expressions like \(\ln(x)\) and integrals involving these logarithms. Here, \(\ln(x)\) represents a function that tells you what power you must raise the base of the natural logarithm (which is \(e\) approximately 2.718) to generate x. Understanding properties of logarithms aids in simplifying integrals:

• *Derivatives:* Recall that the derivative of \(\ln x\) is \(1/x\), a crucial fact when using substitution techniques in integration.
• *Integration with logarithms:* Basic integration with logs might involve integrating functions where \(1/x\) is involved directly, or indirectly through substitutions such as making \(u = \ln x\).

In the context of indefinite integrals, logarithmic functions often allow you to simplify the expression and find an antiderivative, facilitating the overall integration process.

Integration Techniques

Integration techniques are methods used to find integrals, especially when faced with complex expressions that standard rules don't immediately resolve. Mastering these techniques broadens your ability to tackle various types of integrals.

Some common integration techniques to consider are:

• *Substitution:* As described earlier, substitution simplifies the integral by changing variables, often making complex integrals more straightforward to handle. This is particularly useful with trigonometric and logarithmic expressions.
• *Integration by Parts:* A technique that’s useful when dealing with the product of functions. However, for some integrals, like the given example, simpler methods are preferable.
• *Partial Fractions:* Useful in breaking down rational expressions into simpler fractions, which can then be integrated individually.

In practice, the goal is to transform the integral into a form that can be solved with one of these techniques. The choice of method depends on the form of the integrand and recognizing familiar patterns or transformations within it. In the example exercise, substitution was key to solving the integral effectively, highlighting how choosing the right technique streamlines the process.