Solving Problems Involving Repeated Integration by Parts

Introduction

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

1. Identify \( u \) and \( dv \): Choose \( u \) to be a function that simplifies upon differentiation and \( dv \) to be the remaining part of the integrand.
2. Differentiate and Integrate: Compute \( du = \frac{du}{dx} dx \) and \( v = \int dv \).
3. Apply Integration by Parts: Substitute the identified parts into the formula: $$\int u \, dv = uv - \int v \, du$$
4. Assess the New Integral: Determine if the resulting integral requires another application of integration by parts. If so, repeat the process.
5. 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., \( e^x \))

Examples of Repeated Integration by Parts

Example 1: Evaluate \( \int x e^x dx \).

1. Choose \( u = x \) (algebraic) and \( dv = e^x dx \).
2. Differentiate and integrate: $$du = dx$$ $$v = \int e^x dx = e^x$$
3. 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 \( \int x^2 \sin x dx \).

1. 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: $$\int x^2 \sin x dx = -x^2 \cos x + \int 2x \cos x dx$$
2. 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: $$\int 2x \cos x dx = 2x \sin x - \int 2 \sin x dx = 2x \sin x + 2 \cos x + C$$
3. 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

Summary and Key Takeaways