Integration by parts

Definition

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and provides a systematic way to transform the integral of a product into simpler integrals. This method is particularly useful when dealing with the integration of functions that can be decomposed into a product, allowing for easier evaluation.

5 Must Know Facts For Your Next Test

1. The formula for integration by parts is derived from the product rule and is given by $$\int u \, dv = uv - \int v \, du$$, where u and dv are chosen from the integrand.
2. Choosing u and dv wisely is crucial; typically, u should be a function that becomes simpler when differentiated, while dv should be easy to integrate.
3. Integration by parts can sometimes require multiple applications if the resulting integral is still complex.
4. This method can also be applied to definite integrals, but one must evaluate the bounds at the end after performing the integration.
5. Common scenarios where integration by parts is useful include integrals involving logarithmic functions, polynomials multiplied by exponential functions, or trigonometric functions.

Review Questions

• How do you select the appropriate functions for u and dv when using integration by parts?
  • Selecting functions for u and dv is an important step in integration by parts. A common approach is to use the acronym LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. This suggests that you should choose u as the first function from this list that appears in your integral, which helps ensure that the derivative of u will simplify the integral of dv.
• Compare integration by parts with u-substitution. In what situations would one be preferred over the other?
  • Integration by parts and u-substitution are both techniques used in integration, but they apply to different types of integrals. U-substitution is best when an integral can be simplified by changing variables, particularly when there is a clear inner function whose derivative is also present. On the other hand, integration by parts is more effective for integrals involving products of functions or when dealing with logarithmic or exponential functions where direct substitution does not simplify matters.
• Evaluate the integral $$\int x e^x \, dx$$ using integration by parts and discuss how this illustrates the technique's utility.
  • To evaluate $$\int x e^x \, dx$$ using integration by parts, we set u = x (hence du = dx) and dv = e^x \, dx (thus v = e^x). Applying the formula gives us: $$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$$. This example highlights the power of integration by parts since it transforms a more complex integral into simpler components, demonstrating how it can simplify otherwise challenging problems in calculus.

Related terms

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

U-substitution: A technique used in integration that simplifies the process by substituting a part of the integral with a new variable, making it easier to solve.

Definite Integral: An integral that evaluates the area under a curve between two specified limits, providing a numerical result.