Problem 36 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{\cos \sqrt{x}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral is \( 2 \sin(\sqrt{x}) + C \).

Step 1: Identify a Suitable Substitution

Identify a variable substitution to simplify the integral. Let \( u = \sqrt{x} \), then \( du = \frac{1}{2\sqrt{x}}dx \). Solving for \( dx \), we have \( dx = 2u \, du \). Also, \( \sqrt{x} = u \), so the integral becomes \( \int \frac{\cos u}{u} \cdot 2u \, du = 2 \int \cos u \, du \).

Step 2: Integrate with Respect to New Variable

Integrate \( 2 \int \cos u \, du \). The antiderivative of \( \cos u \) is \( \sin u \), so we find the integral: \( 2 \int \cos u \, du = 2 \sin u + C \), where \( C \) is the constant of integration.

Step 3: Substitute Back the Original Variable

Replace \( u \) with the original variable, which gives \( u = \sqrt{x} \). So the integral in terms of the original variable is \( 2 \sin(\sqrt{x}) + C \).

Step 4: Differentiate to Verify Solution

Differentiate \( 2 \sin(\sqrt{x}) + C \) with respect to \( x \) to verify. The derivative of \( 2 \sin(\sqrt{x}) \) is \( 2 \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \).\ The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \), so \( 2 \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} = \frac{\cos(\sqrt{x})}{\sqrt{x}} \). This matches the original integrand \( \frac{\cos \sqrt{x}}{\sqrt{x}} \).

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, which is used to find the area under curves, among other applications. Many functions are straightforward to integrate using basic rules, but others require advanced techniques due to their complexity. For example, the given integral \( \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx \) can't be solved directly with elementary rules. To tackle these complex integrals, mathematicians use various techniques:

• **Substitution**: A method to simplify the function to a basic form which is easier to integrate.
• **Integration by Parts**: Useful when the integral is a product of two functions.
• **Partial Fractions**: Applied when dealing with rational functions.
• **Trigonometric Integrals and Substitutions**: Helps simplify integrals involving trigonometric functions.

Choosing the right technique depends heavily on the function's structure. It's crucial to identify patterns or simplifications that can simplify the integration.

Substitution Method

The substitution method, also known as \( u \)-substitution, is particularly useful for integrals where a component of the integrand can be expressed as a simpler function. For the integral \( \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx \), substitution is beneficial because of the nested function \( \cos \sqrt{x} \).Here's the process:1. **Choose a Substitution:** Let \( u = \sqrt{x} \). This choice breaks down a complex expression into a simpler one.2. **Determine \( du \):** Differentiate \( u = \sqrt{x} \) to find \( du \). Thus, \( du = \frac{1}{2\sqrt{x}}dx \) which rearranges to \( dx = 2u \, du \).3. **Rewrite the Integral:** Substitute back into the integral. It transforms to \( 2 \int \cos u \, du \), a much simpler integral to solve.The substitution method often requires understanding which part of the integrand, when replaced, leads to simplification. Practicing different substitutions increases familiarity and intuition.

Antiderivative Verification

Once you've found an antiderivative, the next step is verification, ensuring it's correct. Differentiation is used to confirm your integral result matches the original function.Taking the derived result, \( 2 \sin(\sqrt{x}) + C \), we differentiate to check:- Differentiate \( 2 \sin(\sqrt{x}) \). Use the chain rule: - Derivative of \( \sin(\sqrt{x}) \) is \( \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \). - \( \frac{d}{dx}(\sqrt{x}) \) is \( \frac{1}{2\sqrt{x}} \). Combining these gives us, \( 2 \cdot \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} = \frac{\cos(\sqrt{x})}{\sqrt{x}} \).This matches the original integrand \( \frac{\cos \sqrt{x}}{\sqrt{x}} \). Thus, our antiderivative is confirmed correct.Antiderivative verification is essential as a final check. It ensures the integration process accurately reverses differentiation back to the original function.