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Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is very useful in simplifying the process of integration for certain types of functions. In the context of our exercise, we needed to complete the square for the expression in the denominator of the integral \[ x^2 + 4x - 5 \].
By rewriting it in terms of a square, we transformed it into \[ (x + 2)^2 - 3^2 \]. This transformation is helpful because it makes the expression resemble a difference of squares, which is easier to handle during integration using standard forms found in integral tables.
To complete the square, follow these steps:

• Take the coefficient of the linear term (half of 4 is 2), square it (2 squared is 4), and add and subtract this number (4 in this case) inside the expression.
• Re-write the quadratic expression as a square of a binomial minus a constant term.

This procedure helps us adopt a form that is more conducive to other integration techniques.

The substitution method makes use of a change of variables to simplify the process of integration. It involves substituting part of the integrand with a single variable to transform the integral into a simpler one. In this exercise, after completing the square with \[ (x + 2)^2 - 3^2 \], we opted for a substitution where \( u = x + 2 \).
Once this substitution was made, the differential becomes \( du = dx \), which transforms our integral into \( \int \frac{1}{u^2 - 3^2} \, du \).This is a much simpler form and is consistent with typical forms found in integral tables.
Substitution:

• Helps transform the integral into a form that is easier to evaluate.
• Is especially effective when the integrand resembles a standard form after a substitution.

Remember, after integrating, you must substitute the original variable back into the solution, as shown in the final step of our exercise.

Integral tables are collections of standard integral forms and their antiderivatives. They serve as a useful tool in calculus to quickly find integrals without performing the integration manually.
In our exercise, the problem was solved by recognizing the transformation \( \int \frac{1}{u^2 - 3^2} \, du \) as a standard form integral. This particular form matched \( \int \frac{1}{u^2 - a^2} \, du \), whose solution is known to be \( \frac{1}{2a} \ln \left| \frac{u-a}{u+a} \right| + C \).
Using integral tables:

• Can save time by providing solutions to frequently encountered forms.
• Is beneficial when the integrand aligns with specific known integrals.

In our case, the integral table provided a seamless transition to the solution post-substitution, greatly simplifying the process.

Computer Algebra Systems (CAS) are software tools used to perform symbolic mathematics, including solving integrals and derivatives. They are powerful because they can handle complex calculations quickly and verify solutions obtained through manual methods.
In our exercise, a CAS was used to confirm the results obtained through square completion and substitution. By inputting \( \int \frac{1}{x^2 + 4x - 5} \, dx \) directly into a CAS, we verified that the software's output matched our manually derived solution \( \frac{1}{6} \ln \left| \frac{x-1}{x+5} \right| + C \).
Benefits of CAS:

• Offer a rapid method for checking calculations and verifying results.
• Provide a reliable way to tackle more complicated integrals.
• Helpful in educational settings to enhance students' understanding of mathematical processes.

Therefore, while hand calculations are valuable for learning, CAS tools offer a practical way to ensure accuracy and efficiency.