Question

Use a table of integrals to evaluate the following integrals. $$ \int \frac{1}{4 x^{2}+25} d x $$

Short answer

\( \frac{1}{5} \arctan\left(\frac{2x}{5}\right) + C \)

Step-by-step solution

Identify the Integral Form

The given integral is \( \int \frac{1}{4x^2 + 25} \, dx \). We need to identify a similar form from the table of integrals that we can use to solve this. Typically, it resembles the integral \( \int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \). We must express the given integral in this form.

Identify Constants

We need to rewrite \( 4x^2 + 25 \) in the form \( a^2 + u^2 \). Notice that \( 4x^2 = (2x)^2 \), and thus the expression can be rewritten as \( (2x)^2 + 5^2 \). This means in our integral, \( a = 5 \) and \( u = 2x \).

Apply the Formula

Using the identified constants, the integral \( \int \frac{1}{a^2 + u^2} \, du \) can be applied directly. Substituting the values, we get: \[ \int \frac{1}{25 + (2x)^2} \, dx = \frac{1}{5} \arctan\left(\frac{2x}{5}\right) + C \].

Simplify the Integral

The final expression after evaluating the integral is \( \frac{1}{5} \arctan\left(\frac{2x}{5}\right) + C \). Thus, the solution to the integral \( \int \frac{1}{4x^2 + 25} \, dx \) is this simplified expression.

Key concepts

Table of Integrals

A table of integrals is a valuable tool in calculus, especially when dealing with complex integrals that don't immediately lend themselves to basic integration techniques. The table consists of a list of standard forms and their corresponding integrals, allowing students to match their given integral to a known formula.
It is essential to become familiar with this table because it can save time and effort when evaluating integrals. By recognizing the form in your problem, you'll know exactly where to look and how to apply the corresponding formula.
The integral in our problem, \( \int \frac{1}{4x^2 + 25} \, dx \), can be found in the table under forms similar to \( \int \frac{1}{a^2 + u^2} \, du \), leading directly to the arctangent solution.

Integration Techniques

Integration techniques are various methods used to solve integrals that cannot be solved directly by basic integration rules. Common techniques include substitution, integration by parts, partial fractions, and trigonometric identities. Each technique has its domain of applicability and understanding when and how to use them is crucial.
In this exercise, the technique involved is recognizing and reformulating the integrand to fit a known integral form from the table of integrals. By rewriting \( 4x^2 + 25 \) as \( (2x)^2 + 5^2 \), we transformed the integral into a standard form. This manipulation allows for the straightforward application of a formula, simplifying the process significantly.

Arctangent Function

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function. In the realm of integration, it frequently appears in integrals involving expressions of the form \( 1/(a^2 + u^2) \).
The significance of the arctangent function is that it helps us express angles and their relationships in integration problems dealing with right triangles and trigonometry.
In our solution, the problem involves the direct use of the formula \( \int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \), leading to a result involving \( \arctan \). The function simplifies what would otherwise be a complex integration process.

Step-by-Step Solutions

Solving integrals step-by-step is crucial for understanding how the various integration techniques are applied. Each step builds upon the previous one, enhancing comprehension for the student.
Let's summarize the process we used in the exercise:

• Identified the similarity between our integral \( \int \frac{1}{4x^2 + 25} \, dx \) and a standard form \( \int \frac{1}{a^2 + u^2} \, du \).
• Re-expressed the integrand carefully by setting \( 4x^2 = (2x)^2 \) and \( 25 = 5^2 \), identifying \( a = 5 \) and \( u = 2x \).
• Applied the corresponding formula from the table of integrals to solve the problem.
• Simplified the result to obtain the final expression, \( \frac{1}{5} \arctan\left(\frac{2x}{5}\right) + C \).

This methodical approach is vital when tackling complex calculus problems, ensuring that no step is skipped and each aspect is fully understood.