Integration is a fundamental concept in calculus, used to find areas, volumes, and solutions to various mathematical problems. One common integral that often appears in calculus problems is the **integral of xe^x **.
In this topic, we will explore **how to integrate xe^x ** using the integration by parts method, explain its step-by-step solution, and discuss its practical applications in mathematics and engineering.
Understanding the Function xe^x
Before integrating, let’s break down the function:
- x is a polynomial term.
- e^x is an exponential function.
When these two terms are multiplied, integration by parts is the best approach to solve the integral.
The Formula for Integration by Parts
Integration by parts is derived from the product rule for differentiation and follows this formula:
Here, we need to select:
- u (a function that simplifies when differentiated).
- dv (a function that is easy to integrate).
For the integral int xe^x dx , the best choices are:
- u = x Rightarrow du = dx
- dv = e^x dx Rightarrow v = e^x
Step-by-Step Solution
Using the integration by parts formula:
Substituting the chosen values:
Since the integral of e^x is simply e^x , we get:
where C is the constant of integration.
Final Answer
Why Does Integration by Parts Work?
The integration by parts method helps in solving products of functions where one function becomes simpler after differentiation. In this case, differentiating x gives 1, reducing the complexity of the integral.
Applications of the Integral of xe^x
1. Physics and Engineering
- Solving differential equations in physics and engineering problems.
- Modeling exponential growth and decay with additional variables.
2. Probability and Statistics
- Used in continuous probability distributions where exponentials appear in density functions.
3. Economics and Finance
- Helps in calculating continuous interest rates and growth models.
Common Mistakes to Avoid
- Choosing u and dv incorrectly
- The polynomial term should be u for easier differentiation.
- **Forgetting the constant C **
- Always include + C in indefinite integrals.
- Skipping integration steps
- Writing each step ensures accuracy.
The integral of xe^x is solved using integration by parts, and the final answer is:
This method is useful for handling integrals involving the product of polynomial and exponential functions. Understanding this concept enhances problem-solving skills in calculus and its applications in various fields.