key term - ∫ x^n e^{ax} dx

Definition

The expression ∫ x^n e^{ax} dx represents the integral of a polynomial function multiplied by an exponential function, where n is a non-negative integer and a is a constant. This type of integral often arises in applications involving growth and decay processes, as well as in solving differential equations. Understanding how to evaluate this integral is crucial since it combines techniques from both polynomial integration and exponential functions.

5 Must Know Facts For Your Next Test

1. The integral ∫ x^n e^{ax} dx can be solved using the method of integration by parts, which may require multiple iterations for higher values of n.
2. In general, the solution to this integral results in a combination of terms involving both polynomials and exponential functions.
3. For n = 0, the integral simplifies to ∫ e^{ax} dx, which has a straightforward solution: (1/a)e^{ax} + C.
4. The result of the integral can be expressed as a sum of terms, each involving decreasing powers of x multiplied by e^{ax}.
5. This integral is particularly useful in solving problems related to differential equations, especially those that model real-world phenomena.

Review Questions

• How does the technique of integration by parts apply when evaluating the integral ∫ x^n e^{ax} dx?
  • When evaluating ∫ x^n e^{ax} dx using integration by parts, you typically let u = x^n and dv = e^{ax} dx. This choice allows you to differentiate u to get du = n x^{n-1} dx while integrating dv results in v = (1/a)e^{ax}. The formula then gives you uv - ∫ v du, leading to an expression that simplifies the original integral. This process often requires repeating integration by parts until reaching an integral that is easily solvable.
• What happens to the integral ∫ x^n e^{ax} dx when n is zero, and how does this special case help in understanding more complex cases?
  • When n is zero, the integral reduces to ∫ e^{ax} dx, which can be directly computed as (1/a)e^{ax} + C. This simple case serves as a foundation for understanding more complex integrals because it highlights how the exponential function behaves on its own. Knowing this base case allows you to better grasp how polynomial factors influence the resulting integral when combined with exponentials.
• Evaluate how the integral ∫ x^n e^{ax} dx relates to real-world applications such as population growth models or radioactive decay.
  • The integral ∫ x^n e^{ax} dx plays a significant role in modeling real-world phenomena like population growth or radioactive decay because these processes often involve exponential changes over time, represented mathematically by functions of the form e^{ax}. By incorporating polynomial factors, this integral allows for more nuanced models that account for varying rates or external influences over time. The resulting solutions from these integrals help predict future states or quantities in such scenarios, making them invaluable in fields like biology and physics.

Related terms

Integration by Parts:

A technique used to integrate products of functions, based on the formula ∫ u dv = uv - ∫ v du.

Exponential Function: A mathematical function of the form e^{ax}, where e is the base of natural logarithms and a is a constant.

Polynomial: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.