integration of the derivative

Oct 13, 2025 by windailys.com 46 views

Introduction

Integration is a fundamental concept in calculus, and it plays a crucial role in solving problems in various fields, including physics, engineering, and economics. In this article, we will focus on evaluating a specific definite integral, $\int_1^2 x e^x \, dx$ . This integral is a classic example of a product of functions, and it requires the use of integration by parts to solve.

Background

Before we dive into the solution, let's briefly review the concept of integration by parts. Integration by parts is a technique used to integrate the product of two functions. It states that if we have two functions, $u$ and $v$ , then the integral of their product can be written as:

\int u \, dv = uv - \int v \, du

This formula allows us to integrate the product of two functions by differentiating one function and integrating the other.

Solution

To evaluate the integral $\int_1^2 x e^x \, dx$ , we will use the integration by parts formula. Let's choose $u = x$ and $dv = e^x \, dx$ . Then, we have:

du = dx \quad \text{and} \quad v = e^x

Now, we can apply the integration by parts formula:

\int_1^2 x e^x \, dx = \left[ x e^x \right]_1^2 - \int_1^2 e^x \, dx

The first term on the right-hand side is the evaluation of the product, and the second term is the integration of the derivative.

Evaluation of the Product

To evaluate the product $x e^x$ at the limits of integration, we need to use the chain rule. The chain rule states that if we have a function of the form $f(g(x))$ , then the derivative is given by:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

In this case, we have $f(g(x)) = x e^x$ , where $g(x) = x$ . Therefore, we can write:

\frac{d}{dx} (x e^x) = e^x + x e^x

Now, we can evaluate the product at the limits of integration:

\left[ x e^x \right]_1^2 = 2 e^2 - e^1

Integration of the Derivative

The second term on the right-hand side is the integration of the derivative. We can integrate the derivative $e^x$ using the exponential integral formula:

\int e^x \, dx = e^x + C

where $C$ is the constant of integration. Therefore, we have:

\int_1^2 e^x \, dx = \left[ e^x \right]_1^2 = e^2 - e^1

Final Answer

Now, we can combine the results from the previous steps to obtain the final answer:

\int_1^2 x e^x \, dx = \left[ x e^x \right]_1^2 - \int_1^2 e^x \, dx = 2 e^2 - e^1 - (e^2 - e^1)

Simplifying the expression, we get:

\int_1^2 x e^x \, dx = e^2 - e^1 + e^1 - e^2 + e^1 = 2 e^1 - e^2

Therefore, the final answer is:

\boxed{\int_1^2 x e^x \, dx = 2 e - e^2}

Conclusion

In this article, we evaluated the definite integral $\int_1^2 x e^x \, dx$ using the integration by parts formula. We chose $u = x$ and $dv = e^x \, dx$ , and then applied the integration by parts formula to obtain the final answer. The result is a classic example of a product of functions, and it requires the use of integration by parts to solve.

References

[1] Calculus, by Michael Spivak
[2] Calculus: Early Transcendentals, by James Stewart
[3] Introduction to Calculus, by Michael Sullivan

Introduction

In our previous article, we evaluated the definite integral $\int_1^2 x e^x \, dx$ using the integration by parts formula. In this article, we will answer some common questions related to this topic.

Q: What is the integration by parts formula?

A: The integration by parts formula is:

\int u \, dv = uv - \int v \, du

This formula allows us to integrate the product of two functions by differentiating one function and integrating the other.

Q: How do I choose the functions $u$ and $dv$ ?

A: To choose the functions $u$ and $dv$ , you need to identify the product of functions that you want to integrate. Then, you can choose one function to be $u$ and the other function to be $dv$ . For example, in the integral $\int_1^2 x e^x \, dx$ , we chose $u = x$ and $dv = e^x \, dx$ .

Q: What is the difference between the product and the derivative?

A: The product is the result of multiplying two functions together, while the derivative is the result of differentiating one function with respect to the other.

Q: How do I evaluate the product at the limits of integration?

A: To evaluate the product at the limits of integration, you need to use the chain rule. The chain rule states that if you have a function of the form $f(g(x))$ , then the derivative is given by:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

In this case, we have $f(g(x)) = x e^x$ , where $g(x) = x$ . Therefore, we can write:

\frac{d}{dx} (x e^x) = e^x + x e^x

Q: What is the exponential integral formula?

A: The exponential integral formula is:

\int e^x \, dx = e^x + C

where $C$ is the constant of integration.

Q: How do I integrate the derivative?

A: To integrate the derivative, you can use the exponential integral formula. For example, in the integral $\int_1^2 e^x \, dx$ , we can integrate the derivative $e^x$ using the exponential integral formula:

\int e^x \, dx = e^x + C

Q: What is the final answer?

A: The final answer is:

\int_1^2 x e^x \, dx = 2 e - e^2

Q: What are some common mistakes to avoid when evaluating definite integrals?

A: Some common mistakes to avoid when evaluating definite integrals include:

Not choosing the correct functions $u$ and $dv$
Not evaluating the product at the limits of integration correctly
Not integrating the derivative correctly
Not simplifying the expression correctly

Q: What are some tips for evaluating definite integrals?

A: Some tips for evaluating definite integrals include:

Choose the correct functions $u$ and $dv$
Evaluate the product at the limits of integration correctly
Integrate the derivative correctly
Simplify the expression correctly
Use the chain rule to evaluate the product
Use the exponential integral formula to integrate the derivative

Conclusion

In this article, we answered some common questions related to evaluating the definite integral $\int_1^2 x e^x \, dx$ . We hope that this article has been helpful in clarifying some of the concepts and techniques involved in evaluating definite integrals.

References

[1] Calculus, by Michael Spivak
[2] Calculus: Early Transcendentals, by James Stewart
[3] Introduction to Calculus, by Michael Sullivan