Integration by parts

Definition

Integration by parts is a technique used to integrate products of functions by transforming the integral into a simpler form. This method is based on the product rule for differentiation and can be especially useful when integrating the product of a polynomial and an exponential, logarithmic, or trigonometric function. It connects various concepts of calculus, such as the computation of areas, properties of definite integrals, and the manipulation of integrals involving special functions.

5 Must Know Facts For Your Next Test

1. The formula for integration by parts is given by $$\int u \ dv = uv - \int v \ du$$, where u and v are functions of x.
2. Choosing the right functions for u and dv is crucial; typically, you want u to be easier to differentiate and dv to be easy to integrate.
3. This technique can be applied multiple times if necessary, particularly when dealing with more complex integrals.
4. Integration by parts is especially effective for integrating products involving logarithmic and trigonometric functions combined with polynomials.
5. The method can also help derive formulas for specific integrals and connect different integration techniques like partial fractions and trigonometric integrals.

Review Questions

• How does the choice of u and dv affect the application of integration by parts?
  • Choosing u and dv wisely is essential for successfully applying integration by parts. Ideally, u should be a function that simplifies when differentiated, while dv should be easily integrable. If this choice is not optimal, it could lead to more complex integrals that may not simplify effectively. This strategy ensures that subsequent calculations remain manageable and lead towards finding a solution.
• Demonstrate how integration by parts can be utilized in finding areas between curves involving polynomial and exponential functions.
  • To find areas between curves where one function is polynomial and another is exponential, integration by parts can be applied effectively. For example, if we need to integrate $$x e^x$$ to find an area, we can set $$u = x$$ and $$dv = e^x dx$$. By using integration by parts, we obtain $$\int x e^x dx = x e^x - \int e^x dx$$. This simplifies our work, allowing us to compute the area between curves much more easily than direct integration would.
• Evaluate how integration by parts can facilitate solving complex integrals that arise in physics or engineering applications.
  • In physics or engineering, complex integrals often involve products of functions like time-dependent variables or physical laws expressed as products. By employing integration by parts, one can break down these complex expressions into simpler components. For instance, calculating work done involves integrating force over distance, which may lead to complicated expressions needing careful manipulation. Utilizing this method not only simplifies these calculations but also enhances our understanding of underlying physical principles by demonstrating relationships among various variables.

Related terms

Product Rule: A rule in calculus used to differentiate products of two functions, stating that the derivative of a product is the first function times the derivative of the second function plus the second function times the derivative of the first.

Definite Integral: An integral that calculates the accumulation of quantities over a specified interval, resulting in a numerical value representing area under a curve.

Antiderivative: A function whose derivative gives the original function, used in integration to find the area under curves and solve differential equations.