Question

Evaluate the following integrals. $$\int \frac{\ln a x}{x} d x, a \neq 0$$

Short answer

Question: Evaluate the integral \(\int \frac{\ln(ax)}{x} dx\). Answer: \(\int \frac{\ln(ax)}{x} dx = (\ln(ax) - C_1) \cdot \ln|x| - \int \frac{\ln |x|}{x} dx + C\), where \(C_1\) and \(C\) are constants.

Step-by-step solution

Identify u and dv

We are given the integral \(\int \frac{\ln(ax)}{x} dx\). To use integration by parts, we let \(u = \ln(ax)\) and \(dv = \frac{1}{x} dx\).

Differentiate u and integrate dv

Differentiate \(u\) with respect to \(x\) to find \(du\): $$ du = \frac{d(\ln(ax))}{dx} = \frac{a}{ax} dx = \frac{1}{x} dx $$ Integrate \(dv\) with respect to \(x\) to find \(v\): $$ v = \int \frac{1}{x} dx = \ln|x| + C_1 $$ where \(C_1\) is an arbitrary constant.

Use integration by parts formula

Now, we can use the integration by parts formula: \(\int u dv = uv - \int v du\). Plugging in the values for \(u\), \(v\), \(du\), and \(dv\), we get: $$ \int \frac{\ln(ax)}{x} dx = \left(\ln(ax)\right)(\ln|x| + C_1) - \int(\ln|x| + C_1) \frac{1}{x} dx $$

Simplify and integrate the remaining integral

Expand the integral: $$ \int \frac{\ln(ax)}{x} dx = \ln(ax) \cdot \ln|x| - C_1\ln(ax) - \int \left(\frac{\ln|x|}{x} + \frac{C_1}{x}\right) dx $$ Now, we need to integrate the remaining terms: $$ \int \left(\frac{\ln a x}{x} + \frac{C_1}{x}\right) dx = \int \frac{\ln |x|}{x} dx + C_1 \int \frac{1}{x}dx $$ Note that \(\int \frac{1}{x} dx = \ln|x| + C_2\), where \(C_2\) is another arbitrary constant. Therefore, $$ \int \left(\frac{\ln a x}{x} + \frac{C_1}{x}\right) dx = \int \frac{\ln |x|}{x} dx + C_1 \ln|x|+ C_2 $$ To find the integral \(\int \frac{\ln |x|}{x} dx\), we can use integration by parts once more, but this time the integral becomes the same as the original one. Therefore, we can't find an elementary function for this integral. So, our answer will be in terms of this integral.

Write the final result

Substitute the result from Step 4 back into the expression from Step 3: $$ \int \frac{\ln(ax)}{x} dx = \ln(ax) \cdot \ln|x| - C_1\ln(ax) - \left(\int \frac{\ln |x|}{x} dx + C_1 \ln|x| + C_2\right) $$ Combine the constants \(C_1\) and \(C_2\) into a single constant \(C\) and simplify: $$ \int \frac{\ln(ax)}{x} dx = (\ln(ax) - C_1) \cdot \ln|x| - \int \frac{\ln |x|}{x} dx + C $$ This is the final result for the given integral.

Key concepts

Logarithmic Integration

Logarithmic integration is a technique that often involves integrals where a natural logarithm term, such as \( \ln(x) \), is present. It becomes particularly useful when using integration by parts, as it allows us to handle different functions that multiply with such logarithmic terms. In the exercise at hand, we encounter the natural logarithm, \( \ln(ax) \), divided by \( x \). By appropriately choosing which part of the expression to differentiate and integrate, logarithmic functions can simplify integrals that otherwise seem challenging. The key is to use integration by parts to break down the function into more manageable parts.Integration by parts relies on the formula:\[ \int u \, dv = uv - \int v \, du \], where you choose \( u \) and \( dv \) strategically. Here, selecting \( u = \ln(ax) \) allows us to easily differentiate it and make the calculation more tractable.

Definite and Indefinite Integrals

Integrals are essential components of calculus, mainly classified as definite and indefinite. An indefinite integral, often denoted by \( \int f(x) \, dx \), represents the general form of the antiderivative function without specific bounds. The result includes a constant, typically written as \( C \), to account for all potential antiderivatives.

• Indefinite integrals represent families of functions, each differing by a constant.
• Definite integrals have set limits of integration, giving a precise area under the curve.

In our exercise, the focus is on indefinite integrals, where solutions generally present one or more constants (e.g., \( C_1, C_2 \)). These constants are valuable because they represent a family of solutions rather than a single curve.When dealing with complex logarithmic expressions, indefinite integrals can appear indefinite without a simple elementary result, highlighting the beauty and complexity of calculus.

Calculus Techniques

Calculus offers various methods to tackle problems, and among them, Integration by Parts is one of the most powerful techniques. It's specifically handy when you deal with complicated product integrals, such as \( \int \frac{\ln(ax)}{x} \, dx \), as seen in the problem.For effective application, always:

• Select \( u \) and \( dv \) so that \( du \) and \( v \) simplify the problem.
• Ensure the choice reduces complexity after applying \( \int u \, dv = uv - \int v \, du \).

Simplifying integrals that are initially complex through parts requires thoughtful selection and sometimes revisiting integration by parts to achieve the solution.Furthermore, combining integration by parts with other calculus techniques, such as logarithmic properties and simplifications, paves the way to solving intricate integrals. This combination forms the core of many calculus problems, ensuring students gain both a fundamental and applied understanding of mathematical principles.