Question

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. $$\int \frac{\ln |x|\arcsin (\ln |x|)}{x}dx$$

Short answer

$\ln |x|\arcsin (\ln |x|)+\sqrt{1-(\ln |x|{)}^2}+C$.

Step-by-step solution

Identify the Integral Form

In this exercise, we are given the integral $\int \frac{\ln |x|\arcsin (\ln |x|)}{x}dx$. We need to transform it to match a standard form available in integration tables. Notice that the integrand involves both a logarithmic and an arc sine function. A common substitution for integrals involving $\ln |x|$ is to let $u=\ln |x|$, which simplifies the problem.

Make the Substitution

Let's substitute $u=\ln |x|$. Then, $du=\frac{1}{x}dx$, which implies that $dx=xdu$. The integral becomes $\int \arcsin (u)du$.

Evaluate the Integral Using the Table

Now, using the integration table, we want to find $\int \arcsin (u)du$. According to integration tables, the integral $\int \arcsin (u)du=u\arcsin (u)+\sqrt{1-u^2}+C$, where $C$ is the constant of integration.

Substitute Back to the Original Variable

Substitute back $u=\ln |x|$ into the equation from Step 3. This gives us:$$\ln |x|\arcsin (\ln |x|)+\sqrt{1-(\ln |x|{)}^2}+C.$$

Finalize the Solution

The final answer for the integral $\int \frac{\ln |x|\arcsin (\ln |x|)}{x}dx$ is:$$\ln |x|\arcsin (\ln |x|)+\sqrt{1-(\ln |x|{)}^2}+C.$$

Key concepts

Substitution Method

The substitution method is a clever technique used in integration to simplify complex integrals. The main idea is to transform the integral into a simpler form that is easier to evaluate. To do this, we introduce a new variable, typically denoted as $u$, in place of a part of the integrand. This process is often called "u-substitution."
Take our example: $\int \frac{\ln |x|\arcsin (\ln |x|)}{x}dx$. We can see that $\ln |x|$ is a good candidate for substitution. By letting $u=\ln |x|$, we transform a part of the integral into just $u$, which simplifies computations.
After the substitution, it's crucial to also find $du$, which in this case is $\frac{1}{x}dx$. This relationship allows us to replace $dx$ in the integral with $xdu$, resulting in an integral in terms of $u$. This reduction is at the heart of why substitution is so useful: it rephrases the problem into a form that may be more easily integrated.

Integration Tables

Integration tables are essential tools for solving integrals. They provide a list of common integrals and their corresponding solutions, eliminating the need to calculate each integral from scratch. When using integration tables, our goal is to match the given integral to one of the forms listed in these tables.
In our exercise, after performing the substitution with $u=\ln |x|$, the integral is simplified to $\int \arcsin (u)du$. By referring to integration tables, we find that:

• $\int \arcsin (u)du=u\arcsin (u)+\sqrt{1-u^2}+C$

Here, $C$ is the constant of integration, a necessary inclusion because integration is an indefinite operation, and the tables account for this by adding $C$ to every antiderivative. Using these tables cuts down on manually computing challenging integrals, especially when an integral’s antiderivative is not straightforward to determine.

Logarithmic Integration

Logarithmic integration involves integrals that include the natural logarithm function, $\ln (x)$. These integrals often require special techniques like substitution to simplify and solve. In our exercise, $\ln |x|$ appears in the integrand.
The natural logarithm function can sometimes make integration tricky due to its unique properties. However, logarithmic functions frequently signal where substitution might benefit us, especially in expressions where $x$ appears both inside a $\ln |x|$ term and elsewhere in the integrand.
The substitution $u=\ln |x|$ transforms the integral into a form devoid of logarithms, simplifying the problem towards an earlier step where standard integration techniques can be easily applied.

Arc Sine Function

The arc sine function, $\arcsin (x)$, is the inverse of the sine function, which means it answers the question: "For what angle is sine equal to $x$?" In integration, functions like $\arcsin (x)$ add complexity to the integrands due to their inverse trigonometric nature.
In our example, $\arcsin (\ln |x|)$ was part of the original integral. After substitution, the integral $\int \arcsin (u)du$ needs to be solved. Thankfully, integration tables provide us with a handy solution:

• $\int \arcsin (u)du=u\arcsin (u)+\sqrt{1-u^2}+C$

By following the steps and substituting back $u=\ln |x|$, we convert the integral of $\arcsin (u)$ back into terms of $x$, ensuring our solution is complete and more meaningful with respect to the original problem.