key term - ∫ x e^{ax} dx

Definition

The expression ∫ x e^{ax} dx represents the integral of the product of a polynomial term, x, and an exponential function, e^{ax}. This integral is significant as it showcases how to integrate functions involving both polynomial and exponential components, which often appear in real-world applications such as physics and engineering.

5 Must Know Facts For Your Next Test

1. To solve ∫ x e^{ax} dx, you typically use integration by parts, which is based on the formula ∫ u dv = uv - ∫ v du.
2. In this integral, you can choose u = x and dv = e^{ax} dx, making it straightforward to differentiate and integrate respectively.
3. The result of ∫ x e^{ax} dx leads to a combination of polynomial terms and exponential functions in its final expression.
4. The final answer for ∫ x e^{ax} dx can be expressed as (x e^{ax})/a - (e^{ax})/a^2 + C, where C is the constant of integration.
5. Understanding this integral is essential for solving more complex problems involving differential equations and applied mathematics.

Review Questions

• How does integration by parts apply when solving the integral ∫ x e^{ax} dx?
  • Integration by parts is crucial for solving the integral ∫ x e^{ax} dx because it allows us to break down the product of the polynomial and exponential functions. By selecting u = x and dv = e^{ax} dx, we can differentiate u to get du = dx and integrate dv to obtain v = (1/a)e^{ax}. This method results in the formula ∫ u dv = uv - ∫ v du, leading us to find the solution effectively.
• Discuss the steps involved in finding the integral ∫ x e^{ax} dx using integration by parts.
  • To find ∫ x e^{ax} dx using integration by parts, start by letting u = x and dv = e^{ax} dx. Differentiate u to get du = dx and integrate dv to obtain v = (1/a)e^{ax}. Next, apply the integration by parts formula: ∫ u dv = uv - ∫ v du. This gives you (x e^{ax})/a - ∫ (e^{ax}/a) dx. Solving this remaining integral will yield the final answer for ∫ x e^{ax} dx.
• Evaluate the implications of mastering the integral ∫ x e^{ax} dx for understanding more advanced calculus concepts.
  • Mastering the integral ∫ x e^{ax} dx has significant implications for understanding advanced calculus concepts, particularly in applications such as differential equations and modeling real-world scenarios. This integral exemplifies how polynomial and exponential functions interact under integration, providing foundational skills that are applicable in fields like physics and engineering. Furthermore, a solid grasp of integration techniques like integration by parts enhances problem-solving abilities, enabling students to tackle more complex integrals with confidence.

Related terms

Integration by Parts:

A technique used to integrate products of functions, which is essential for solving integrals like ∫ x e^{ax} dx.

Exponential Function: A mathematical function of the form e^{ax}, where 'a' is a constant, known for its rapid growth and unique properties in calculus.

Polynomial: An algebraic expression that consists of variables and coefficients, often represented in the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.