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.
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To solve ∫ x e^{ax} dx, you typically use integration by parts, which is based on the formula ∫ u dv = uv - ∫ v du.
In this integral, you can choose u = x and dv = e^{ax} dx, making it straightforward to differentiate and integrate respectively.
The result of ∫ x e^{ax} dx leads to a combination of polynomial terms and exponential functions in its final expression.
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.
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.
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.