Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{\ln x}{x}dx$$

Short answer

The indefinite integral is $\frac{(\ln x{)}^2}{2}+C$.

Step-by-step solution

Recognize the Need for Substitution

Identify the function and the part of it that can be substituted to simplify the integral. In this case, notice that the presence of $\ln x$ in the numerator and $x$ in the denominator suggests a substitution method can be applied.

Perform the Substitution

Choose $u=\ln x$. Then, compute the derivative of $u$, which gives $du=\frac{1}{x}dx$. This implies that $dx=xdu$. Substitute these into the integral, giving $\int udu$.

Integrate the Substituted Function

Now, integrate $\int udu$. Using the power rule for integration, this integral becomes $\frac{u^2}{2}+C$, where $C$ is the constant of integration.

Substitute Back to Original Variable

Replace $u$ back with $\ln x$ to return to the original variable's terms. Thus, $\frac{u^2}{2}+C$ becomes $\frac{(\ln x{)}^2}{2}+C$.

Key concepts

Substitution Method

The substitution method is a powerful tool in calculus, especially for solving integrals. It simplifies complex integrals by transforming them into more manageable forms. In our example, we need to find the indefinite integral of $\int \frac{\ln x}{x}dx$. This setup is perfect for using substitution since the derivative of $\ln x$ is $\frac{1}{x}$, conveniently matching the denominator.

To apply the substitution method, we start by choosing a variable to substitute. We set $u=\ln x$ and calculate its differential, which is $du=\frac{1}{x}dx$. This immense simplification allows us to change the variable in the integral from $x$ to $u$, thus transforming $\int \frac{\ln x}{x}dx$ into $\int udu$.

This new integral is significantly easier to handle. After integrating, remember to substitute the original variable back into the result. This completes the substitution method, turning complex integrals into simpler ones by temporarily switching variables.

Integration Techniques

Integration techniques are essential tools for solving different types of integrals. They allow us to handle integrals that might not be straightforward at first glance. In this context, using substitution is one such technique. It is especially useful when the integral has a structure that matches a known derivative.

In our exercise, after substituting $u=\ln x$, we integrate $\int udu$. This requires the application of a basic integration rule: the power rule. The power rule states that $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$ for any real number $neq-1$. Here, $n=1$, so we get $\int udu=\frac{u^2}{2}+C$.

This technique simplifies the work, allowing us to solve the integral without directly dealing with the original complex functions. It's an elegant way to break down daunting problems into manageable parts using standard integration rules.

Logarithmic Functions

Logarithmic functions, like $\ln x$, have unique properties that make them interesting and occasionally challenging in integration. They often appear in integrals where substitution is beneficial. Understanding the behavior of logarithmic functions, especially their derivatives, is crucial.

In the integral $\int \frac{\ln x}{x}dx$, the expression involves the natural logarithm $\ln x$. One key property of logarithmic functions is that their derivative, $\frac{d}{dx}(\ln x)$, is $\frac{1}{x}$. This property is what makes substitution possible here and transforms the integral into a simpler form.

When dealing with logarithmic functions in integrals, always check if their derivatives might help simplify the problem. This knowledge not only helps reduce complicated calculus challenges but also deepens your understanding of the behavior and characteristics of logarithmic functions.