Integration by Parts

Definition

Integration by parts is a technique used in calculus to integrate the product of two functions. This method is based on the product rule for differentiation and transforms the integral of a product into simpler integrals, making it easier to evaluate complex integrals. The technique is often applied when one of the functions in the product can be easily differentiated while the other can be easily integrated.

5 Must Know Facts For Your Next Test

1. The integration by parts formula is given by $$\int u \; dv = uv - \int v \; du$$, where u and v are differentiable functions.
2. Choosing u and dv appropriately is key to successfully applying integration by parts, as it can significantly affect the complexity of the resulting integrals.
3. It is often useful to apply integration by parts multiple times when integrating functions that yield more complex products after differentiation.
4. This technique is particularly useful for integrating products involving polynomial and exponential functions or trigonometric functions.
5. Integration by parts can also help in evaluating definite integrals when combined with limits, ensuring proper application of boundary conditions.

Review Questions

• How does the choice of u and dv impact the outcome of using integration by parts?
  • The choice of u and dv directly influences the simplicity and solvability of the resulting integrals when using integration by parts. A good choice typically involves selecting u to be a function that simplifies upon differentiation, while dv should be a function that can be easily integrated. If chosen poorly, the process can lead to more complex integrals that may require additional methods or steps to evaluate.
• Discuss how integration by parts can be applied to definite integrals and how limits are handled in this context.
  • When applying integration by parts to definite integrals, it is essential to evaluate the boundaries after finding the antiderivative. The formula remains the same: $$\int_a^b u \; dv = [uv]_a^b - \int_a^b v \; du$$. Here, you first compute $$uv$$ at the limits a and b, and then subtract the integral of $$v \; du$$ over those same limits. This ensures that both the indefinite and definite aspects are properly addressed during evaluation.
• Evaluate how integration by parts compares with substitution method in terms of effectiveness for different types of integrals.
  • Integration by parts and substitution are both valuable techniques in calculus, but they serve different purposes based on the functions involved. Substitution is typically more effective for integrals involving composite functions where one function clearly defines a simpler variable. In contrast, integration by parts shines when dealing with products of functions where differentiating one component simplifies the overall expression. Depending on the complexity and form of the integral, one method may yield results faster or more straightforwardly than the other.

Related terms

Product Rule: A rule in calculus that provides a formula for the derivative of the product of two functions.

Definite Integral: An integral that calculates the accumulation of a quantity, defined over a specific interval.

Substitution Method: A technique used to simplify an integral by substituting a part of the integral with a single variable.