Question

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers. $$ \int \frac{d x}{x \sqrt{x-1}} $$

Short answer

\( 2 \arctan(\sqrt{x-1}) + C \).

Step-by-step solution

Analyze the Integral

First, look at the integral \( \int \frac{d x}{x \sqrt{x-1}} \). The integrand has the form \( \frac{1}{x \sqrt{x-1}} \), which suggests a possible substitution or transformation might be beneficial due to the presence of a square root.

Substitution

Notice that the expression \( \sqrt{x-1} \) suggests a substitution to simplify the integral. Try the substitution \( x = u^2 + 1 \) so that \( dx = 2u \, du \) and \( \sqrt{x-1} = u \). The integral becomes \( \int \frac{2u \, du}{(u^2 + 1)u} = \int \frac{2 \, du}{u^2 + 1} \).

Integrate using CAS

Using a CAS (Computer Algebra System), compute \( \int \frac{2}{u^2 + 1} \, du \). This integral is a standard form and evaluates to \( 2 \arctan(u) + C \).

Reverse Substitution

Replace \( u \) back in terms of \( x \) using the substitution \( u = \sqrt{x-1} \). Thus, the integral becomes \( 2 \arctan(\sqrt{x-1}) + C \).

Verify with Integral Tables

Consult an integral table to check the result. The integral \( \int \frac{d x}{x \sqrt{x-1}} \) is indeed solved as \( 2 \arctan(\sqrt{x-1}) + C \), confirming the solution found using CAS.

Key concepts

Substitution Method

The substitution method is a powerful technique in integral calculus that simplifies complex integrals. This is done by changing the variable of integration to make the integral easier to solve. For the given integral, \( \int \frac{dx}{x \sqrt{x-1}} \), the presence of the square root suggests that substitution would be beneficial.

To simplify, we substitute \( x = u^2 + 1 \), which transforms the expression \( \sqrt{x-1} \) directly into \( u \). This substitution alters the differential: \( dx = 2u \, du \). By making these changes, our integral simplifies to \( \int \frac{2u \, du}{(u^2 + 1)u} \), which further reduces to \( \int \frac{2 \, du}{u^2 + 1} \). This substitution method allows us to work with more manageable expressions, making the integration process more straightforward.

Substitution makes the integration of otherwise challenging functions possible by converting them into standard forms whose antiderivatives are known or easier to compute.

Reverse Substitution

Once the integration is performed in terms of the new variable, it's crucial to return to the original variable. This process is known as reverse substitution. After completing the integral with the substitution method in our example, we obtained \( 2 \arctan(u) + C \).

Reverse substitution involves replacing \( u \) back with \( \sqrt{x-1} \), as per our original substitution \( x = u^2 + 1 \). Consequently, the final expression is \( 2 \arctan(\sqrt{x-1}) + C \).

Reverse substitution is essential to ensure that the solution's context remains consistent with the original problem. Forgetting to perform reverse substitution is a common mistake, but regular practice helps build confidence in completing this step accurately.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) can be an invaluable tool in solving integrals, especially when they are complex. CAS software leverages symbolic computation to automatically perform algebraic operations, such as finding the integral of a function.

In our example, after reaching the integral \( \int \frac{2}{u^2 + 1} \, du \), a CAS quickly computed the solution \( 2 \arctan(u) + C \). One advantage of using CAS is that it reduces human error and saves time on complex calculations.

Moreover, a CAS can verify solutions and provide alternative forms of an answer, offering deeper insights into the behavior of integrals. For students, learning to use a CAS effectively can enhance their problem-solving skills and complement their understanding of manual calculation techniques.