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The u-substitution method is a technique used in calculus to simplify integration by making a substitution. It involves substituting part of the integrand with a new variable to make the integral more manageable. This method is particularly useful when dealing with complex expressions in the integrand that resemble standard integral forms.
In the exercise provided, we identify the part of the expression that, when substituted, simplifies the integral. Here, we notice the term \(4x^2\), which can be simplified by using the substitution \(u = 2x\). As a result, the differential \(du\) becomes \(2 \, dx\), or equivalently, \(dx = \frac{1}{2} \, du\).
The key steps in performing u-substitution in this example are:

• Recognize a part of the integrand that matches a standard form when substituted.
• Express \(dx\) in terms of \(du\) to substitute it into the integral.
• Rewrite the integral using the u-substitution to arrive at a simpler form.

Applying u-substitution transforms the integral into a format that is easier to integrate, thus facilitating the solution process.

The integration involving the arctangent function is often encountered in expressions that resemble the standard integral form \( \int \frac{1}{a^2 + u^2} \, du \). The result of such an integral is given by \( \frac{1}{a} \arctan\left( \frac{u}{a} \right) + C \).
This particular exercise uses arctangent integration once the expression is substituted using a u-substitution. After substituting \(u = 2x\) in the original problem, the integral simplifies to \(\frac{1}{2} \int \frac{1}{9 + u^2} \, du\). In this form, it directly matches the standard form for arctangent integration with \(a = 3\).
Some key points to remember with arctangent integration:

• Recognize the form \(a^2 + u^2\) in the denominator, which signifies the use of arctangent integration.
• Perform u-substitution first if needed, to match this form.
• Apply the formula directly to evaluate the integral.

This process helps convert the integral into a simpler process, enhancing the understanding of how trigonometric functions can simplify calculus problems.

Definite integrals are used to calculate the area under a curve between two specific limits. However, the exercise in question does not specify limits for a definite integral; it instead focuses on the indefinite integral, concluding with the inclusion of the constant \(C\).
In contexts where definite integrals are specified, you would follow similar steps for integration using u-substitution and arctangent methods, but with one key difference: you would evaluate the resulting function at the upper and lower limits.
If a definite integral had been given, the resulting steps would include:

• Performing the same integration process using u-substitution and arctangent integration.
• Evaluating the integrated expression using the limits specified in the problem.
• Subtracting the value of the function at the lower limit from the value at the upper limit to find the desired area.

Understanding definite integrals is essential in calculus as it relates to real-world applications like calculating distances and areas. However, this exercise helps build foundational skills with indefinite integrals which are stepping stones to mastering definite integrals.