Question

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x \ln x} d x $$

Short answer

\(\ln|\ln x| + C\)

Step-by-step solution

Identify the Substitution

The denominator \(x \ln x\) suggests that we should use the substitution \(u = \ln x\). This causes the original integral to take a more familiar form that can be easily handled.

Compute the Differential dx

Differentiating \(u\) with respect to \(x\), we have:\(du/dx = 1/x\),which implies that \(dx = x du\). This will replace \(dx\) in the original integral.

Substitute 'u' and 'dx' in the Integral

Substitute \(u\) and \(dx\) in the integral and rewrite it as:\(\int \frac{1}{xu} \cdot x du = \int \frac{1}{u} du\). Now, the integral is easier to compute.

Evaluate the Integral

The integral of \(1/u\) with respect to \(u\) is \(\ln|u|\). So, the integral becomes \(\ln|u|+C\).

Back-Substitute \(u = \ln x\)

Replace \(u\) with \(\ln x\) to get the final answer:\(\ln|\ln x|+C\).

Key concepts

Integration Techniques

Integration is one of the fundamental operations in calculus, alongside differentiation. When faced with a function that needs to be integrated, various integration techniques can be employed. These techniques are strategies to simplify or transform integrals into a more manageable form for which the antiderivative can be more easily found.

Some common integration techniques include:

• Substitution Method (u-substitution): This process involves replacing a function or a portion of the function with a new variable to simplify the integral.
• Integration by Parts: Useful especially when the integrand is a product of functions. It’s based on the product rule of differentiation.
• Partial Fractions: This method decomposes a complex rational function into simpler fractions that can be integrated individually.
• Trigonometric Substitution: It exploits the properties of trigonometric identities to simplify integrands involving roots.

The choice of the technique is usually determined by the form of the integrand and sometimes requires trial and error or the familiarity with specific integral forms.

Logarithmic Integration

Logarithmic integration is an integral part of calculus and is often employed when an integrand involves logarithmic functions. The rules of integration that involve logarithms can be intricate, but one important one to remember is that the integral of the function \(1/u\) with respect to \(u\) is the natural logarithm of the absolute value of \(u\), notated as \(\ln|u| + C\), where \(C\) represents the constant of integration.

To integrate functions involving logarithms effectively, you should:

• Recognize patterns where logarithms can be taken advantage of, such as functions of the form \(\frac{1}{g(x)}g'(x)\).
• Know the derivatives and integrals of logarithmic functions like \(\frac{d}{dx}\ln|x| = \frac{1}{x}\) and the aforementioned integral \(\int \frac{1}{u} du = \ln|u| + C\).

By understanding these rules, you can tackle a variety of integrals that otherwise might seem daunting due to the presence of logarithmic expressions.

U-Substitution

U-substitution is a powerful technique in calculus used to find the integrals of more complex functions. It is akin to the chain rule for differentiation but in reverse. When you encounter an integral that isn't straightforward to compute, like \(\int \frac{1}{x \ln x} dx\), using u-substitution can often simplify the process.

Steps Involved in U-Substitution:

• Identify a part of the integrand to substitute with a variable \(u\), usually chosen to simplify the integral or to make it resemble a known integral form.
• Express \(dx\) in terms of \(du\) by differentiating \(u\) with respect to \(x\).
• Substitute \(u\) and \(dx\) into the integral, simplifying where possible.
• Perform the integration with respect to \(u\).
• Substitute the original variable back in place of \(u\) to obtain the result in terms of the original variable.

By effectively using this substitution method, even functions that initially appear complex can be integrated successfully.