Repeated integration by parts is a fundamental technique in calculus, particularly within the subject of Calculus BC under the Collegeboard AP curriculum. This method extends the basic integration by parts formula to solve more complex integrals that cannot be addressed through a single application. Mastery of this technique is essential for students aiming to excel in advanced calculus topics and standardized examinations.

Key Concepts

Understanding Integration by Parts

Integration by parts is a method derived from the product rule for differentiation. It is used to integrate products of functions and is formally stated as: $$\int u \, dv = uv - \int v \, du$$ where $ u $ and $ dv $ are continuously differentiable functions of $ x $.

The Need for Repeated Integration by Parts

Some integrals require more than one application of the integration by parts formula to be solved. This typically occurs when the resulting integral after the first application is still complex or similar to the original integral. Repeated integration by parts often leads to a solvable equation after rearranging terms, especially when dealing with polynomial and exponential functions multiplied by trigonometric functions.

Step-by-Step Procedure

Identify $ u $ and $ dv $: Choose $ u $ to be a function that simplifies upon differentiation and $ dv $ to be the remaining part of the integrand.
Differentiate and Integrate: Compute $ du = \frac{du}{dx} dx $ and $ v = \int dv $.
Apply Integration by Parts: Substitute the identified parts into the formula: $$\int u \, dv = uv - \int v \, du$$
Assess the New Integral: Determine if the resulting integral requires another application of integration by parts. If so, repeat the process.
Solve for the Original Integral: After sufficient repetitions, arrange all instances of the original integral on one side of the equation and solve algebraically.

Choosing $ u $ and $ dv $: LIATE Rule

The selection of $ u $ and $ dv $ is crucial for simplifying the integral. The LIATE rule provides a hierarchy for choosing $ u $:

L: Logarithmic functions (e.g., $ \ln x $)
I: Inverse trigonometric functions (e.g., $ \arctan x $)
A: Algebraic functions (e.g., $ x^n $)
T: Trigonometric functions (e.g., $ \sin x $)
E: Exponential functions (e.g., $ e^x $)

Choosing $ u $ based on this hierarchy often leads to simpler integrals after differentiation and integration steps.

Examples of Repeated Integration by Parts

Example 1: Evaluate $ \int x e^x dx $.

Choose $ u = x $ (algebraic) and $ dv = e^x dx $.
Differentiate and integrate: $$du = dx$$ $$v = \int e^x dx = e^x$$
Apply integration by parts: $$\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$$

Example 2: Evaluate $ \int x^2 \sin x dx $.

First Integration by Parts:
- Choose $ u = x^2 $ and $ dv = \sin x dx $.
- Differentiate and integrate: $$du = 2x dx$$ $$v = -\cos x$$
- Apply the formula: $$\int x^2 \sin x dx = -x^2 \cos x + \int 2x \cos x dx$$
Second Integration by Parts on $ \int 2x \cos x dx $:
- Choose $ u = 2x $ and $ dv = \cos x dx $.
- Differentiate and integrate: $$du = 2 dx$$ $$v = \sin x$$
- Apply the formula: $$\int 2x \cos x dx = 2x \sin x - \int 2 \sin x dx = 2x \sin x + 2 \cos x + C$$
Combine the results: $$\int x^2 \sin x dx = -x^2 \cos x + 2x \sin x + 2 \cos x + C$$

Handling Reduction Formulas

In cases where repeated integration by parts leads to a series of integrals, reduction formulas can be utilized to express the original integral in terms of a simpler one. This is particularly useful for integrals involving powers of $ x $ multiplied by trigonometric or exponential functions.

Solving for the Integral

After applying integration by parts multiple times, the original integral often reappears on both sides of the equation. By algebraically rearranging the terms, the integral can be solved for explicitly. This method ensures that the integral is expressed in terms of known functions without further integration requirements.

Common Mistakes to Avoid

Incorrect selection of $ u $ and $ dv $, leading to more complicated integrals.
Failing to recognize when to stop applying integration by parts.
Algebraic errors during the rearrangement of terms.
Neglecting to include the constant of integration $ C $ in indefinite integrals.

Practice Problems

To solidify understanding, students should practice solving integrals that require repeated integration by parts:

Evaluate $ \int x^3 e^x dx $.
Find $ \int \ln x dx $.
Compute $ \int x^2 \cos x dx $.

Applications in Real-World Problems

Repeated integration by parts is not only a theoretical exercise but also has practical applications in physics and engineering, such as in calculating work done by varying forces, moments of inertia, and solving differential equations that model real-world phenomena.

Comparison Table

Aspect	Basic Integration by Parts	Repeated Integration by Parts
Definition	Applies the formula $ \int u \, dv = uv - \int v \, du $ once.	Applies the integration by parts formula multiple times to solve complex integrals.
Complexity	Suitable for integrals where one application suffices.	Necessary for integrals that remain complex after the first application.
Typical Use Cases	Integrals involving a single product of algebraic and exponential/trigonometric functions.	Integrals involving higher powers of polynomials multiplied by exponential or trigonometric functions.
Pros	Simple and straightforward when applicable.	Expands the applicability of integration by parts to a broader range of integrals.
Cons	Limited to simpler integrals.	Can be time-consuming and requires careful application to avoid errors.

Summary and Key Takeaways

Repeated integration by parts extends the basic method to solve more complex integrals.
Proper selection of $ u $ and $ dv $ using the LIATE rule simplifies the process.
Algebraic manipulation is essential to solve for the original integral after multiple applications.
Practice with diverse problems enhances proficiency and accuracy in applying the technique.
This method has practical applications in various scientific and engineering fields.

Examiner Tip

Tips

Use the LIATE rule as a mnemonic to select $ u $ and $ dv $ effectively: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Practice breaking down complex integrals into simpler parts and always double-check your differentiation and integration steps. For AP exam success, solve a variety of problems under timed conditions to build confidence and speed.

Did You Know

Integration by parts dates back to Isaac Newton and Gottfried Wilhelm Leibniz, the founders of calculus. Surprisingly, repeated integration by parts can be used to derive the Taylor series for functions like sine and cosine. Additionally, this technique plays a crucial role in solving differential equations that model real-world phenomena such as oscillations and heat transfer.

Common Mistakes

One frequent error is choosing $ dv $ incorrectly, leading to more complex integrals. For example, selecting $ u = e^x $ and $ dv = x dx $ in $ \int x e^x dx $ complicates the process, whereas using the LIATE rule suggests $ u = x $ and $ dv = e^x dx $. Another mistake is forgetting to include the constant of integration $ C $, which is essential in indefinite integrals.

FAQ

When should I use repeated integration by parts?

Use repeated integration by parts when the first application leaves you with an integral similar to the original or when dealing with products of polynomials and exponential or trigonometric functions.

What is the LIATE rule?

The LIATE rule helps in choosing $ u $ and $ dv $ by prioritizing Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions.

Can all integrals be solved using integration by parts?

No, integration by parts is effective for specific types of integrals, especially those involving products of functions. Some integrals require other techniques like substitution or partial fractions.

How do I recognize when to stop applying integration by parts?

Stop when the integral becomes straightforward to solve or when the original integral reappears in the equation, allowing you to solve for it algebraically.

What are common pitfalls to avoid?

Avoid selecting $ u $ and $ dv $ that complicate the integral, ensure accurate differentiation and integration, and always include the constant of integration for indefinite integrals.

How does repeated integration by parts apply to real-world problems?

It is used in physics and engineering for calculating work, moments of inertia, and solving differential equations that model phenomena like oscillations and heat transfer.

1. Integration and Accumulation of Change

1.1 Evaluating Improper Integrals

1.1.1 Evaluating Improper Integrals Using Limits

1.1.2 Determining Convergence or Divergence of Improper Integrals

1.1.3 Understanding Improper Integrals with Infinite Limits

1.2 Integration by Parts

1.2.1 Understanding the Integration by Parts Formula

1.2.2 Choosing u and dv for Integration by Parts

1.2.3 Solving Problems Involving Repeated Integration by Parts

1.3 Using Linear Partial Fractions

1.3.1 Decomposing Rational Functions into Partial Fractions

1.3.2 Solving Integrals Using Partial Fraction Decomposition