Question

Evaluate the definite integral \(\int_{0}^{2} \frac{d x}{4+x^{2}}\).

Short answer

The value of the integral is \( \frac{\pi}{8} \).

Step-by-step solution

Identify the Integral Type

The given integral is \( \int_{0}^{2} \frac{d x}{4+x^{2}} \). This is a standard integral that resembles the form \( \int \frac{d x}{a^2+x^2} \), which is related to the inverse tangent function.

Apply the Integral Formula

The formula for integration is \( \int \frac{d x}{a^2+x^2} = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C \). Here, \( a^2 = 4 \), so \( a = 2 \). Substitute into the formula to get \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \).

Evaluate the Definite Integral

Convert the indefinite integral into a definite integral: \( \int_{0}^{2} \frac{d x}{4+x^{2}} = \left[ \frac{1}{2} \arctan \left( \frac{x}{2} \right) \right]_{0}^{2} \). Evaluate this expression at the upper and lower limits.

Calculate at the Upper Limit

Substitute \( x = 2 \) into \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \). This gives \( \frac{1}{2} \arctan(1) = \frac{1}{2} \left( \frac{\pi}{4} \right) = \frac{\pi}{8} \).

Calculate at the Lower Limit

Substitute \( x = 0 \) into \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \). This gives \( \frac{1}{2} \arctan(0) = 0 \).

Compute the Result

Subtract the result at the lower limit from the result at the upper limit: \( \frac{\pi}{8} - 0 = \frac{\pi}{8} \).

Conclude the Solution

The value of the definite integral \( \int_{0}^{2} \frac{d x}{4+x^{2}} \) is \( \frac{\pi}{8} \).

Key concepts

Inverse Tangent Function

The inverse tangent function, often represented as \( \arctan(x) \), is a crucial component in the calculus realm, particularly when dealing with specific types of integrals. It defines the angle whose tangent is a given number and serves as the inverse of the tangent function. This function is integral in scenarios where we encounter expressions of the form \( \frac{1}{a^2 + x^2} \) in integrals.

In this exercise, the inverse tangent function becomes essential as we are tasked with integrating \( \int \frac{d x}{4+x^{2}} \). Recognizing the expression as one that involves the inverse tangent is the key step that leads us to apply the right integration formula. Essentially, the presence of this function guides us in resolving the integral by transforming the expression into a more manageable form through substitution.

Integration Formula

Integration formulas are like tool kits for calculus, allowing us to simplify complex expressions. In our integral \( \int \frac{d x}{4+x^{2}} \), the integration formula we use is \( \int \frac{d x}{a^2+x^2} = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C \). This special formula is utilized when the integrand takes on specific types of structures like the one we encounter here.

In our problem, since \( a^2 = 4 \), we find \( a = 2 \). This allows us to effortlessly apply the integration formula by substituting \( a \) directly. After substitution, we get \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \). This transformation simplifies our initial problem significantly and enables us to find the definite integral from the given limits.

Upper and Lower Limits

Upper and lower limits play a crucial role when evaluating definite integrals. They define the range over which we integrate the function. In this instance, our limits are from \( 0 \) to \( 2 \). Once we have determined the appropriate antiderivative through integration – here, using \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \) – we need to assess it at these specific boundary values.

We substitute \( x = 2 \) (the upper limit) and \( x = 0 \) (the lower limit) one by one into our antiderivative. These evaluations provide the respective outputs needed in the next step to finalize the computation of the definite integral. This step ensures we're calculating the precise area under the curve from the lower limit to the upper limit, as defined by the integral boundaries.

Evaluate Definite Integral

Evaluating a definite integral involves calculating the antiderivative at the given upper limit, subtracting the value of the antiderivative at the lower limit. This process grants the actual value of the integral over the specified interval. In our problem, the antiderivative is \( \frac{1}{2} \arctan \left( \frac{x}{2} \right) \).

By plugging \( x = 2 \) into the antiderivative, we compute \( \frac{1}{2} \arctan(1) \), which equals \( \frac{\pi}{8} \). For \( x = 0 \), we find \( \frac{1}{2} \arctan(0) = 0 \). The final step requires subtracting these results, upper minus lower: \( \frac{\pi}{8} - 0 \), yielding \( \frac{\pi}{8} \).

This calculated result embodies the signed area under the curve from \( x = 0 \) to \( x = 2 \), fully solving the integral. This ability to evaluate definite integrals is fundamental for a wide range of applications in mathematical analysis and real-world problem-solving.