Problem 31 - Calculus Single Variable

Question

Use the method of substitution to evaluate the definite integrals. $$ \int_{\pi^{2} / 4}^{\pi^{2}} \frac{\cos (\sqrt{x})}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to -2.

Step 1: Choose the Substitution

To solve the integral \( \int_{\pi^{2} / 4}^{\pi^{2}} \frac{\cos(\sqrt{x})}{\sqrt{x}} \ dx \) using substitution, let's set \( u = \sqrt{x} \). Consequently, \( x = u^2 \) and \( dx = 2u \, du \). This substitution simplifies the integral by removing the square root in the denominator.

Step 2: Determine the New Limits

After substituting \( u = \sqrt{x} \), the limits of integration change. When \( x = \pi^2/4 \), \( u = \pi/2 \). For \( x = \pi^2 \), \( u = \pi \). Hence, the new limits of integration for \( u \) are from \( \pi/2 \) to \( \pi \).

Step 3: Substitute and Simplify the Integral

Apply the substitution in the integral: \( \int_{\pi/2}^{\pi} \frac{\cos(u)}{u} \cdot 2u \, du \). Simplifying, this becomes \( 2 \int_{\pi/2}^{\pi} \cos(u) \, du \).

Step 4: Evaluate the New Integral

The integral \( 2 \int_{\pi/2}^{\pi} \cos(u) \, du \) evaluates to twice the antiderivative of \( \cos(u) \), which is \( \sin(u) \). So, \( 2[\sin(u)]_{\pi/2}^{\pi} = 2(\sin(\pi) - \sin(\pi/2)) = 2(0 - 1) = -2 \).

Step 5: Conclude with the Final Result

Therefore, the definite integral \( \int_{\pi^{2} / 4}^{\pi^{2}} \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx = -2 \).

Key Concepts

Definite Integrals

A definite integral represents the area under a curve within specific limits on a graph. It calculates the total accumulated quantity, like area or displacement, over an interval. When we have a definite integral like \[ \int_{a}^{b} f(x) \, dx\]- where \( a \) and \( b \) are the limits of integration - we're looking for the area bounded between the curve of \( f(x) \) and the x-axis from \( x = a \) to \( x = b \). This concept is foundational in calculus and helps to solve real-world problems involving continuous data.

• Lower and Upper Limits: These are the points \( a \) and \( b \) which define the start and end of the interval over which you're finding the integral.
• Function: \( f(x) \) represents the function graph whose area under the curve you're trying to determine.
• Definite vs. Indefinite Integrals: Unlike indefinite integrals that determine a family of functions, definite integrals provide a specific numerical value as a result.

Understanding how to evaluate these integrals is crucial in fields like physics and engineering. It's about finding how much quantity accumulates over a segment.

Change of Variables

Change of variables, or substitution, is a technique used to simplify the process of integration. By changing the variable, we can turn a complicated integral into a simpler one. This technique involves substituting parts of an integral with a new variable to clean up the expression.

• The Basic Idea: Usually, we start by identifying a part of the original integral that can be replaced with a new variable. In our example, we chose \( u = \sqrt{x} \).
• Substitution Adjustment: Once the variable is replaced, we adjust the differential to fit the substitution. In the case of \( x = u^2 \), we found that \( dx = 2u \, du \).
• Change of Limits: When making a substitution in a definite integral, the limits of integration must also change based on our new substitution. This requires plugging in the original limits into the substitution equation to find new limits.

Using substitution is akin to swapping one piece of a puzzle with another that fits better. It makes solving integrals more straightforward and manageable, as seen when transforming and evaluating the integral in this exercise.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, especially calculus. They describe relationships between angles and sides of triangles, and they play a pivotal role in periodic functions and wavelike phenomena.

• Cosine Function: In our integral, the cosine function \( \cos(\sqrt{x}) \) is integral to the form and solution. \( \cos(x) \) expresses the x-coordinate projection of a point rotating around a circle.
• Periodicity: These functions repeat at regular intervals. For cosine, the period is \( 2\pi \), meaning it returns to the same value every \( 2\pi \) radians.
• Derivative and Antiderivative: The derivative of \( \cos(x) \) is \(-\sin(x)\), and the antiderivative is \( \sin(x) \), both valuable for solving integrals.

Understanding trigonometric functions is essential for interpreting waves, oscillations, and rotations. They are an indispensable part of many scientific and engineering disciplines, providing the tools to model and analyze cyclical patterns.