Verifying Solutions to Antiderivative Problems

Introduction

Key Concepts

Understanding Antiderivatives

An antiderivative of a function $f(x)$ is a function $F(x)$ such that:

$$F'(x) = f(x)$$

Essentially, finding an antiderivative involves reversing the process of differentiation. Antiderivatives are also known as indefinite integrals and are fundamental in defining the accumulation of quantities.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation and integration, providing a method to evaluate definite integrals. It consists of two parts:

1. First Part: If $F(x)$ is an antiderivative of $f(x)$ on an interval $[a, b]$, then:
$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$
2. Second Part: If $F(x)$ is defined by: $$F(x) = \int_{a}^{x} f(t) dt$$ then $F'(x) = f(x)$.

This theorem not only allows for the evaluation of definite integrals but also provides a means to verify antiderivatives.

Techniques for Finding Antiderivatives

Various techniques are employed to find antiderivatives, each suited to different types of functions:

• Basic Integration: Involves straightforward application of integration rules, such as power rule, exponential functions, and trigonometric functions.
• Integration by Substitution: Utilizes a change of variable to simplify the integral, often used when dealing with composite functions.
• Integration by Parts: Based on the product rule for differentiation, this method is useful for integrating products of functions.
• Partial Fraction Decomposition: Breaks down rational functions into simpler fractions that are easier to integrate.

Verification Methods for Antiderivatives

After finding an antiderivative, it is crucial to verify its correctness. The most common method involves differentiation:

• Differentiation Check: Differentiate the proposed antiderivative and confirm that it equals the original function $f(x)$. If $F'(x) = f(x)$, then $F(x)$ is a valid antiderivative.
• Comparison with Known Antiderivatives: Compare the obtained antiderivative with standard integrals to ensure consistency.
• Graphical Verification: Analyze the graphs of $F(x)$ and $f(x)$ to observe the relationship between a function and its antiderivative.

Common Antiderivatives and Their Verifications

Below are some common antiderivatives along with their verification processes:

Power Functions

For $f(x) = x^n$, where $n \neq -1$, an antiderivative is:

$$F(x) = \frac{x^{n+1}}{n+1} + C$$

Verification: Differentiate $F(x)$:

$$F'(x) = \frac{d}{dx}\left(\frac{x^{n+1}}{n+1} + C\right) = x^n$$

Since $F'(x) = f(x)$, the antiderivative is verified.

Exponential Functions

For $f(x) = e^{kx}$, an antiderivative is:

$$F(x) = \frac{e^{kx}}{k} + C$$

Verification: Differentiate $F(x)$:

$$F'(x) = \frac{d}{dx}\left(\frac{e^{kx}}{k} + C\right) = e^{kx}$$

Thus, $F'(x) = f(x)$, confirming the antiderivative.

Trigonometric Functions

For $f(x) = \cos(kx)$, an antiderivative is:

$$F(x) = \frac{\sin(kx)}{k} + C$$

Verification: Differentiate $F(x)$:

$$F'(x) = \frac{d}{dx}\left(\frac{\sin(kx)}{k} + C\right) = \cos(kx)$$

The equality $F'(x) = f(x)$ verifies the antiderivative.

Step-by-Step Example

Problem: Find an antiderivative of $f(x) = 3x^2 + 2x - 5$ and verify it.

Solution:

1. Find the Antiderivative:

Integrate each term separately:

$$\int 3x^2 dx = 3 \cdot \frac{x^3}{3} = x^3$$ $$\int 2x dx = 2 \cdot \frac{x^2}{2} = x^2$$ $$\int (-5) dx = -5x$$

Combine the results:

$$F(x) = x^3 + x^2 - 5x + C$$
2. Verify the Antiderivative:

Differentiate $F(x)$:

$$F'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(x^2) - \frac{d}{dx}(5x) + \frac{d}{dx}(C)$$ $$F'(x) = 3x^2 + 2x - 5$$

Since $F'(x) = f(x)$, the antiderivative is verified.

Integration by Substitution

Integration by substitution is a powerful technique used when an integral contains a composite function. The goal is to simplify the integral by making a substitution that reduces it to a basic form.

Example: Find an antiderivative of $f(x) = (2x) \cdot e^{x^2}$.

Solution:

1. Choose a substitution: Let $u = x^2$.
2. Differentiate: $du/dx = 2x$, so $du = 2x dx$.
3. Rewrite the integral in terms of $u$: $$\int (2x) e^{x^2} dx = \int e^{u} du$$
4. Integrate: $$\int e^{u} du = e^{u} + C = e^{x^2} + C$$

Verification: Differentiate $F(x) = e^{x^2} + C$:

$$F'(x) = 2x e^{x^2}$$

Which matches $f(x) = 2x e^{x^2}$, thereby verifying the antiderivative.

Integration by Parts

Integration by parts is based on the product rule for differentiation and is used to integrate products of functions.

Formula:

$$\int u \, dv = uv - \int v \, du$$

Example: Find an antiderivative of $f(x) = x \cdot \ln(x)$.

Solution:

1. Choose $u$ and $dv$:
   • Let $u = \ln(x)$, so $du = \frac{1}{x} dx$.
   • Let $dv = x dx$, so $v = \frac{x^2}{2}$.
2. Apply Integration by Parts: $$\int x \ln(x) dx = \frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} dx$$ Simplify the integral: $$= \frac{x^2}{2} \ln(x) - \frac{1}{2} \int x dx$$ $$= \frac{x^2}{2} \ln(x) - \frac{1}{2} \cdot \frac{x^2}{2} + C$$ $$= \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C$$

Verification: Differentiate $F(x) = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C$:

$$F'(x) = \frac{2x}{2} \ln(x) + \frac{x^2}{2} \cdot \frac{1}{x} - \frac{2x}{4}$$ $$F'(x) = x \ln(x) + \frac{x}{2} - \frac{x}{2}$$ $$F'(x) = x \ln(x)$$

Thus, $F'(x) = f(x)$ verifies the antiderivative.

Partial Fraction Decomposition

Partial fraction decomposition is used to integrate rational functions by expressing them as a sum of simpler fractions.

Example: Find an antiderivative of $f(x) = \frac{3x + 5}{x^2 + x}$.

Solution:

1. Factor the denominator: $$x^2 + x = x(x + 1)$$
2. Express as partial fractions: $$\frac{3x + 5}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1}$$
3. Find constants $A$ and $B$: $$3x + 5 = A(x + 1) + Bx$$ Expand and collect like terms: $$3x + 5 = (A + B)x + A$$ Equate coefficients:
   • For $x$: $A + B = 3$
   • Constant term: $A = 5$
   Solving, $A = 5$ and $B = -2$.
4. Rewrite the integral: $$\int \frac{3x + 5}{x(x + 1)} dx = \int \left(\frac{5}{x} - \frac{2}{x + 1}\right) dx$$
5. Integrate: $$\int \frac{5}{x} dx - \int \frac{2}{x + 1} dx = 5 \ln|x| - 2 \ln|x + 1| + C$$

Verification: Differentiate $F(x) = 5 \ln|x| - 2 \ln|x + 1| + C$:

$$F'(x) = \frac{5}{x} - \frac{2}{x + 1}$$ $$F'(x) = \frac{5(x + 1) - 2x}{x(x + 1)}$$ $$F'(x) = \frac{5x + 5 - 2x}{x(x + 1)}$$ $$F'(x) = \frac{3x + 5}{x(x + 1)}$$

Thus, $F'(x) = f(x)$ confirms the antiderivative.

Numerical Methods for Verification

In cases where analytical verification is challenging, numerical methods can be employed:

• Approximation: Use numerical differentiation to approximate $F'(x)$ and compare it with $f(x)$.
• Graphical Analysis: Plot both $F'(x)$ and $f(x)$ to visually inspect their agreement.

Common Mistakes in Verification

Ensuring correctness during verification is vital. Common mistakes include:

• Ignoring the Constant of Integration: Always include the constant $C$ when finding antiderivatives.
• Mistakes in Differentiation: Careful application of differentiation rules is necessary to avoid errors.
• Incorrect Substitutions: In substitution methods, improper choice of $u$ can lead to incorrect results.
• Sign Errors: Pay attention to signs, especially when dealing with negative coefficients.

Advanced Verification Techniques

For more complex functions, advanced techniques may be required:

• Multiple Substitutions: Applying substitution more than once to simplify the integral.
• Trigonometric Identities: Utilize identities to transform and simplify integrands.
• Series Expansion: Expand functions into series and integrate term-by-term when applicable.

The Role of Verification in Problem-Solving

Verification serves multiple purposes in calculus:

• Ensures Accuracy: Confirms that the antiderivative is correct, preventing propagation of errors.
• Builds Understanding: Deepens comprehension of integration techniques and their applications.
• Enhances Confidence: Reinforces the reliability of solutions, boosting problem-solving confidence.

Practice Problems

Engaging with practice problems enhances proficiency in verifying antiderivatives. Below are sample problems for practice:

Problem 1: Find an antiderivative of $f(x) = 4x^3 - 2x + 7$ and verify it.

Solution:

1. Integrate each term: $$\int 4x^3 dx = 4 \cdot \frac{x^4}{4} = x^4$$ $$\int (-2x) dx = -2 \cdot \frac{x^2}{2} = -x^2$$ $$\int 7 dx = 7x$$ $$F(x) = x^4 - x^2 + 7x + C$$
2. Verify by differentiation: $$F'(x) = 4x^3 - 2x + 7$$

   Which matches $f(x)$.

Problem 2: Find an antiderivative of $f(x) = \frac{5}{x}$ and verify it.

Solution:

1. Integrate: $$\int \frac{5}{x} dx = 5 \ln|x| + C$$
2. Verify by differentiation: $$F'(x) = \frac{5}{x}$$

   Which equals $f(x)$.

Problem 3: Find an antiderivative of $f(x) = e^{3x}$ and verify it.

Solution:

1. Integrate: $$\int e^{3x} dx = \frac{e^{3x}}{3} + C$$
2. Verify by differentiation: $$F'(x) = \frac{3e^{3x}}{3} = e^{3x}$$

   Which matches $f(x)$.

Comparison Table

Technique	Definition	Applications	Pros	Cons
Basic Integration	Direct application of standard integration rules.	Polynomials, exponential, and trigonometric functions.	Simple and straightforward.	Limited to basic functions.
Integration by Substitution	Changing variables to simplify the integral.	Composite functions and products involving derivatives of inner functions.	Efficient for many complex integrals.	Choice of substitution may not be obvious.
Integration by Parts	Based on the product rule for differentiation.	Products of polynomials and logarithmic/trigonometric functions.	Handles integrals that are products of different types of functions.	Can lead to more complicated integrals.
Partial Fraction Decomposition	Breaking down rational functions into simpler fractions.	Rational functions where the degree of numerator is less than the denominator.	Simplifies complex rational integrals.	Requires factoring of polynomials, which may be difficult.
Numerical Methods	Approximating integrals using numerical techniques.	Functions that do not have elementary antiderivatives.	Applicable to a wide range of functions.	Provides approximate results, not exact.

Summary and Key Takeaways