Problem 29 - Calculus Single Variable

Question

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{0}^{\pi / 4} \sec ^{2}(x) d x $$

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Identify the Antiderivative

The integrand is \(\sec^2(x)\), and we need to find its antiderivative. The antiderivative of \(\sec^2(x)\) is \(\tan(x)\). This is because the derivative of \(\tan(x)\) is \(\sec^2(x)\).

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus part 2 states that if \( F\) is an antiderivative of \( f\), then \[ \int_a^b f(x)\,dx = F(b) - F(a). \] Therefore, our definite integral becomes \(\tan(x)\) evaluated from \(0\) to \(\frac{\pi}{4}\).

Step 3: Evaluate at the Upper Limit

Substitute the upper limit into the antiderivative: \(\tan\left(\frac{\pi}{4}\right) = 1\). This is because the tangent of \(\frac{\pi}{4}\) radians (or 45 degrees) is 1.

Step 4: Evaluate at the Lower Limit

Substitute the lower limit into the antiderivative: \(\tan(0) = 0\). The tangent of 0 radians is 0.

Step 5: Calculate the Definite Integral

Subtract the evaluation of the lower limit from the upper limit: \[ \tan\left(\frac{\pi}{4}\right) - \tan(0) = 1 - 0 = 1. \] Therefore, the value of the integral is \(1\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differential and integral calculus. It shows how we can compute the definite integral of a function if we know one of its antiderivatives. This theorem has two main parts.

• The first part establishes the relationship between differentiation and integration, asserting that these operations are inverses of each other.
• The second part, which concerns us here, states that if a function is continuous on a closed interval \[a, b\], and \(F\) is an antiderivative of \(f\) over this interval, then the integral of \(f(x)\) from \(a\) to \(b\) is \(F(b) - F(a)\).

When you have a definite integral, this theorem lets you evaluate it by first finding an antiderivative of the integrand. Then, simply subtract the antiderivative evaluated at the lower limit from its value at the upper limit. This technique simplifies solving problems that otherwise would be complicated to compute directly.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. This means if your task is to find the antiderivative of a function \(f(x)\), you need to determine a function \(F(x)\) such that \(F'(x) = f(x)\).
In our exercise, the integrand is \(\sec^2(x)\). The task is then to find a function \(F(x)\) such that its derivative gives \(\sec^2(x)\). This can be challenging if you are unfamiliar with common antiderivatives.
However, in this case, the antiderivative of \(\sec^2(x)\) is \(\tan(x)\), because the derivative of \(\tan(x)\) is \(\sec^2(x)\). Recognizing patterns like this makes solving integrals much easier. Knowing a library of basic derivatives and their corresponding antiderivatives speeds up the evaluation of integrals significantly.

Evaluation of Integrals

The evaluation of integrals is the process of finding the total accumulation of quantities over an interval. For definite integrals, this involves applying the antiderivative you have found and using the limits of integration to get a numerical result.
In our case, the definite integral is \(\int_{0}^{\frac{\pi}{4}} \sec^2(x) \, dx\).

• Substitute the upper limit: Evaluate \(\tan\left(\frac{\pi}{4}\right)\), which equals \(1\) because the tangent of \(\frac{\pi}{4}\) radians is 1.
• Substitute the lower limit: Evaluate \(\tan(0)\), which equals \(0\) because the tangent of \(0\) is 0.
• Subtract these values: The calculation is then \(1 - 0 = 1\).

Thus, the definite integral evaluates to \(1\). Through practice, handling evaluations becomes a straightforward step in dealing with definite integrals.