Question

Find an antiderivative of \(\int \frac{1}{1+4 x^{2}} d x\).

Short answer

The antiderivative is \( \frac{1}{2} \arctan(2x) + C \).

Step-by-step solution

Identify the Integral

We need to find an antiderivative of the expression \( \int \frac{1}{1+4x^2} \, dx \). Notice that it resembles the derivative form of the inverse tangent function, but has a coefficient in front of \( x^2 \).

Rewrite the Integral

Observe that the denominator has the expression \( 1 + (2x)^2 \), which can be associated with the derivative of the arctangent function. Rewrite it as \( \int \frac{1}{1+(2x)^2} \, dx \).

Handle Constant Multiplication

Recognize that \( \int \frac{1}{1+(2x)^2} \, dx = \frac{1}{2} \int \frac{1}{1+y^2} \, dy \) using the substitution \( y = 2x \), hence \( dy = 2 \, dx \).

Apply the Antiderivative of Arctangent

Recall that \( \int \frac{1}{1+y^2} \, dy = \arctan(y) + C \). Substitute back \( y = 2x \) to obtain \( \frac{1}{2} \arctan(2x) + C \).

Write the Antiderivative with Constant

Thus, the antiderivative of the given function is \( \frac{1}{2} \arctan(2x) + C \), where \( C \) is the constant of integration.

Key concepts

Inverse Trigonometric Function

Inverse trigonometric functions are essential tools in calculus, especially when working with integrals like the one given in the original exercise. These functions are the inverses of the standard trigonometric functions, such as sine, cosine, and tangent. They allow us to find angles when given a trigonometric ratio.

• The function we work with here is the inverse tangent, or \( \arctan(x) \). It is used because the integral \( \int \frac{1}{1+x^2} \, dx \) corresponds exactly to this particular inverse trigonometric function.
• When identifying an antiderivative that involves expressions of the form \( 1 + (cx)^2 \), we often relate it to the inverse tangent function. Recognizing this pattern is key to simplifying and solving the integral in steps.
• Understanding the derivative forms of inverse trigonometric functions helps unravel more complex integrals by aligning existing forms within them. This pattern recognition can simplify the process significantly.

Never underestimate the value of these functions as they often provide elegant solutions to integrals that initially seem complicated.

Substitution Method

The substitution method is a powerful technique in calculus used to simplify the integration process. It works by changing variables to make the integral easier to evaluate.

• In this exercise, we see substitution in action when the substitution \( y = 2x \) is made. This change simplifies the original integral expression by transforming \( \frac{1}{1+(2x)^2} \) into \( \frac{1}{1+y^2} \).
• After making the substitution, the integral becomes simpler and matches the form of the inverse tangent function's derivative, allowing us to use known antiderivatives effectively.
• This technique is particularly useful when integration becomes too complex with the variables in their original form. It allows for a strategic simplification.

Whenever you encounter an integral that resembles a known derivative form, consider using substitution to bring it closer to that simpler form. By changing variables strategically, you align the problem with easily solvable patterns.

Integration Techniques

Integration techniques are a crucial aspect of calculus, providing methods to solve a wide array of integral problems. This problem illustrates two common techniques: substitution and recognizing inverse trigonometric forms.

• Recognizing patterns is one technique. In this exercise, we notice the \( 1 + 4x^2 \) in the denominator resembles the form associated with an inverse tangent's derivative, indicating the use of an inverse trigonometric function.
• We've also employed substitution, a fundamental technique often used in tandem with other methods, like pattern recognition. It simplifies the expression to align with known antiderivative forms.
• Combining these techniques, the problem becomes more accessible, and we transition from seeing a complex integral to a solvable one by traditional calculus means.

Effective integration requires familiarity with these techniques. They empower you to identify shortcuts, simplify computations, and approach integrals from a strategic angle. Consider these techniques as tools in your mathematical toolkit.