Verifying Antiderivative Solutions

Introduction

Key Concepts

Understanding Antiderivatives

An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that \( F'(x) = f(x) \). Essentially, finding an antiderivative is the reverse process of differentiation. Antiderivatives are central to solving indefinite integrals, which represent a family of functions differing by a constant.

Basic Integration Rules

To verify antiderivatives, it's essential to be well-versed in basic integration rules, including:

• Power Rule: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \)
• Constant Multiple Rule: \( \int k \cdot f(x) dx = k \int f(x) dx \)
• Sum Rule: \( \int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx \)

Techniques of Integration

Various techniques aid in finding antiderivatives of more complex functions:

• Substitution: Useful when an integral contains a function and its derivative. For example, to integrate \( \int 2x e^{x^2} dx \), let \( u = x^2 \), then \( du = 2x dx \).
• Integration by Parts: Based on the product rule for differentiation, the formula is \( \int u \, dv = uv - \int v \, du \). It's particularly useful for integrating products of polynomials and exponential or trigonometric functions.
• Partial Fractions: Decomposes rational functions into simpler fractions, making them easier to integrate.

Verification Process

Once an antiderivative \( F(x) \) is proposed for \( f(x) \), verification involves differentiating \( F(x) \) and confirming that \( F'(x) = f(x) \). This process ensures the correctness of the integration process and the antiderivative found.

Example: Verify that \( F(x) = \frac{1}{3}x^3 + C \) is an antiderivative of \( f(x) = x^2 \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}\left(\frac{1}{3}x^3 + C\right) = x^2 $$ Since \( F'(x) = f(x) \), the verification is successful.

Applications of Antiderivative Verification

Verifying antiderivatives is crucial in various applications:

• Solving Differential Equations: Ensures solutions to differential equations are accurate.
• Calculating Areas Under Curves: Confirms the antiderivative used in the Fundamental Theorem of Calculus is correct.
• Physics and Engineering: Validates models involving motion, growth, and decay where integration is applied.

Common Mistakes in Verification

Students often encounter pitfalls when verifying antiderivatives:

• Miscalculating Derivatives: Incorrect differentiation leads to false verification results.
• Ignoring the Constant of Integration: Omitting the constant \( C \) can cause discrepancies in verification.
• Application of Incorrect Integration Techniques: Using inappropriate methods for certain functions can result in incorrect antiderivatives.

Advanced Verification Techniques

For more complex functions, advanced techniques may be necessary:

• Using Series Expansions: Expanding functions into series can facilitate term-by-term differentiation for verification.
• Numerical Verification: Approximating derivatives at specific points to confirm antiderivative accuracy.
• Symbolic Computation Tools: Employing software like Mathematica or MATLAB to verify derivatives.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges differentiation and integration. It consists of two parts:

• First Part: If \( f \) is continuous on \([a, b]\) and \( F \) is an antiderivative of \( f \), then: $$ \int_a^b f(x) dx = F(b) - F(a) $$
• Second Part: If \( F(x) \) is defined as: $$ F(x) = \int_a^x f(t) dt $$ and \( f \) is continuous on \([a, b]\), then \( F \) is differentiable and \( F'(x) = f(x) \).

Understanding this theorem is essential for verifying antiderivatives within definite integrals.

Practical Examples

Applying verification through examples enhances comprehension:

Example 1: Verify the antiderivative \( F(x) = \sin(x) + C \) for \( f(x) = \cos(x) \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}[\sin(x) + C] = \cos(x) $$ Since \( F'(x) = f(x) \), the verification is successful.

Example 2: Verify \( F(x) = e^{2x}/2 + C \) for \( f(x) = e^{2x} \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}\left(\frac{e^{2x}}{2} + C\right) = \frac{2e^{2x}}{2} = e^{2x} $$ Thus, \( F'(x) = f(x) \), confirming the antiderivative.

Example 3: Verify \( F(x) = \ln|x| + C \) for \( f(x) = 1/x \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}[\ln|x| + C] = \frac{1}{x} $$ Hence, \( F'(x) = f(x) \), validating the antiderivative.

Integrating Trigonometric Functions

Verification becomes slightly more involved with trigonometric functions:

Example: Verify that \( F(x) = -\cos(x) + C \) is an antiderivative of \( f(x) = \sin(x) \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}[-\cos(x) + C] = \sin(x) $$ Thus, \( F'(x) = f(x) \), confirming correctness.

Integrating Exponential Functions

Exponential functions often require attention to bases during verification:

Example: Verify that \( F(x) = e^{3x}/3 + C \) is an antiderivative of \( f(x) = e^{3x} \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}\left(\frac{e^{3x}}{3} + C\right) = \frac{3e^{3x}}{3} = e^{3x} $$ Since \( F'(x) = f(x) \), the antiderivative is verified.

Integrating Polynomial Functions

Polynomial functions follow straightforward differentiation for verification:

Example: Verify \( F(x) = \frac{2}{5}x^{5/2} + C \) for \( f(x) = 2x^{3/2} \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}\left(\frac{2}{5}x^{5/2} + C\right) = 2x^{3/2} $$ Thus, \( F'(x) = f(x) \), confirming correctness.

Verification with Constants

Incorporating constants requires careful differentiation:

Example: Verify \( F(x) = 4x + C \) for \( f(x) = 4 \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}(4x + C) = 4 $$ Hence, \( F'(x) = f(x) \), confirming the antiderivative.

Handling Absolute Values

When dealing with absolute values, ensure correct differentiation:

Example: Verify \( F(x) = |x| + C \) for \( f(x) = \frac{x}{|x|} \) when \( x \neq 0 \).

Differentiate \( F(x) \): $$ F'(x) = \frac{d}{dx}|x| = \frac{x}{|x|} $$ Thus, \( F'(x) = f(x) \), verifying the antiderivative.

Integrating Rational Functions

Rational functions often require partial fraction decomposition before verification:

Example: Verify \( F(x) = \ln|x| - \ln|x + 1| + C \) for \( f(x) = \frac{1}{x} - \frac{1}{x+1} \).

Differentiate \( F(x) \): $$ F'(x) = \frac{1}{x} - \frac{1}{x+1} $$ Hence, \( F'(x) = f(x) \), confirming the antiderivative.

Piecewise Functions

For piecewise functions, verify each piece separately:

Example: Verify \( F(x) = \begin{cases} x^2 + C & \text{if } x < 1 \\ \sqrt{x} + C & \text{if } x \geq 1 \end{cases} \) for \( f(x) = \begin{cases} 2x & \text{if } x < 1 \\ \frac{1}{2\sqrt{x}} & \text{if } x \geq 1 \end{cases} \).

Differentiate each piece: $$ F'(x) = \begin{cases} 2x & \text{if } x < 1 \\ \frac{1}{2\sqrt{x}} & \text{if } x \geq 1 \end{cases} $$ Thus, \( F'(x) = f(x) \) for all \( x \), verifying the antiderivative.

Comparison Table

Summary and Key Takeaways