Techniques of Integration

Integration by Parts

Integration by parts is a fundamental technique for evaluating integrals involving the product of two functions, especially when simpler methods such as substitution or basic power rules are not applicable. This method is derived from the product rule for differentiation and is essential for solving a wide range of business calculus problems.

• Product of Functions: When integrating a product of two functions, and substitution does not work, integration by parts is often the preferred method.

• Connection to Product Rule: The formula for integration by parts is derived by reversing the product rule for differentiation.

Derivation of the Integration by Parts Formula

• Product Rule for Derivatives: If f(x) and g(x) are differentiable functions, then:

• Integrating Both Sides: Integrate both sides with respect to x:

• Applying the Fundamental Theorem of Calculus: The left side simplifies to f(x)g(x):

• Solving for the Integral of the Product:

Standard Notation for Integration by Parts

• Let u = f(x) and dv = g'(x)dx. Then, du = f'(x)dx and v = g(x).

• The integration by parts formula becomes:

Choosing u and dv

• Guideline: Choose u so that its derivative du is simpler than u itself.

• Choose dv: Select dv as the part of the integrand that is easily integrable.

• Common Choices: Polynomials are often chosen for u, while exponential and trigonometric functions are good candidates for dv.

Example: Integrating

Evaluate using integration by parts.

• Step 1: Choose u and dv

  • Let (since its derivative is simpler: )

  • Let (since it is easy to integrate: )

• Step 2: Apply the Formula

  • Substitute the values:

• Final Answer:

Summary of Steps for Integration by Parts

1. Identify the parts of the integrand to assign to u and dv.

2. Compute du and v by differentiating u and integrating dv.

3. Substitute into the formula .

4. Simplify and integrate the remaining integral.

5. Add the constant of integration C for indefinite integrals.

Key Points

• Integration by parts is especially useful when integrating products of polynomials and exponentials, polynomials and trigonometric functions, or logarithmic functions.

• Choosing u and dv wisely is crucial for simplifying the problem.

• The method is based on reversing the product rule for derivatives.

Additional info: In more advanced cases, integration by parts may need to be applied more than once, or in combination with other integration techniques. Practice with a variety of examples is recommended to master the method.