Question

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} $$

Short answer

The approximate value of the integral is 0.69677571.

Step-by-step solution

Set Up the Definite Integral

The problem is to evaluate \( \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} \). Observing this integral, it has a composite function under the square root, which hints at the use of a substitution or direct evaluation using a Computer Algebra System (CAS).

Using a Computer Algebra System (CAS)

Input the integral into a CAS to find the exact solution. Use the integral bounds from 1 to 3 and the function \( \frac{1}{u \sqrt{2u-1}} \). The CAS computes the indefinite integral and then applies the limits to provide the definite integral value.

Determine Result Format

Check if the CAS provides an exact answer in terms of elementary functions. If not, proceed with the numerical approximation.

Obtain Numerical Approximation

If the CAS does not yield an exact symbolic solution, obtain a numerical approximation. Numerical evaluation using a CAS provides \( \approx 0.69677571 \).

Conclusion

With the steps and CAS output in mind, the definite integral evaluates approximately to \( 0.69677571 \). This is its numerical result when no exact symbolic expression is found.

Key concepts

Computer Algebra System (CAS)

A Computer Algebra System, or CAS, is a powerful tool often used in mathematics, physics, and engineering to perform algebraic manipulations and calculus operations. These programs can handle everything from solving equations to computing integrals and derivatives with precision.

When dealing with complex integrals like \( \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} \), a CAS is incredibly helpful. With just a few inputs: the function, the integration limits, and the requirement for an exact or numerical solution, the CAS does the heavy lifting. It computes indefinite integrals and applies the fundamental theorem of calculus to evaluate definite ones.

The beauty of using a CAS lies in its speed and accuracy. It can quickly determine if a symbolic answer exists in the realm of elementary functions. If a symbolic answer is obscure or not possible, it will switch to providing a numerical approximation.

Numerical Approximation

Numerical approximation comes into play when a definite integral doesn't yield a result in terms of elementary functions. While elementary function solutions are exact, they aren't always available for every integral, particularly those involving complex or nested functions.

In the case of \( \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} \), if a CAS cannot express the result in a simple closed form, it resorts to numerical methods.

These methods utilize mathematical techniques like the trapezoidal rule or Simpson's rule to approximate the area under the curve represented by the integral. Such techniques divide the area into manageable parts and sum up these areas for an estimate. The result given by the CAS, approximately 0.69677571, reflects a sophisticated numerical technique applied quickly and efficiently, aiding in practical applications where an exact number isn't critical.

Elementary Functions

Elementary functions refer to the basic functions in mathematics such as polynomials, exponential functions, logarithms, trigonometric functions, and their inverses. These functions are the building blocks of more complex expressions and are vital in calculus and algebra.

The crux of finding a definite integral like \( \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} \) is determining whether it can be expressed using these basic functions. If it can, we say it has an elementary integral.

Not all functions integrate neatly into elementary forms. In those cases, the integral might involve special functions or be left for numerical methods to approximate. The challenge often involves recognizing patterns or transformations within the integral that match elementary forms, thus simplifying the problem. However, in many practical scenarios, an exact elementary function form might not exist, leading to numerical approximations.