accumulation of a quantity

Oct 13, 2025 by windailys.com 128 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 the definite integral $\int_1^2 x e^x \, dx$ . This integral is a classic example of a trigonometric integral, and it requires the use of various techniques, including integration by parts and substitution.

Background

Before we dive into the evaluation of the integral, let's briefly discuss the concept of integration. Integration is the process of finding the area under a curve or the accumulation of a quantity over a given interval. In this case, we are interested in finding the area under the curve of the function $f(x) = x e^x$ between the limits $x = 1$ and $x = 2$ .

Integration by Parts

One of the most common techniques used to evaluate integrals is integration by parts. This technique is based on the product rule of differentiation, which states that if $u$ and $v$ are two functions of $x$ , then the derivative of their product is given by:

\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}

We can use this formula to evaluate the integral $\int_1^2 x e^x \, dx$ by choosing $u = x$ and $v = e^x$ . Then, we have:

\frac{du}{dx} = 1 \quad \text{and} \quad \frac{dv}{dx} = e^x

Substituting these values into the formula, we get:

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

Now, we can integrate both sides of the equation with respect to $x$ :

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

Using the fundamental theorem of calculus, we can evaluate the left-hand side of the equation:

\int \frac{d}{dx} (uv) \, dx = uv \bigg|_{x=1}^{x=2}

Substituting the values of $u$ and $v$ , we get:

uv \bigg|_{x=1}^{x=2} = x e^x \bigg|_{x=1}^{x=2}

Now, we can evaluate the right-hand side of the equation:

\int (x e^x + e^x) \, dx = \int x e^x \, dx + \int e^x \, dx

Using the integration by parts formula, we can evaluate the first integral:

\int x e^x \, dx = x e^x - \int e^x \, dx

Substituting this value into the previous equation, we get:

\int (x e^x + e^x) \, dx = x e^x - \int e^x \, dx + \int e^x \, dx

Simplifying the equation, we get:

\int (x e^x + e^x) \, dx = x e^x

Now, we can evaluate the integral $\int_1^2 x e^x \, dx$ by substituting the values of $x$ :

\int_1^2 x e^x \, dx = x e^x \bigg|_{x=1}^{x=2}

Evaluating the expression, we get:

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

Conclusion

In this article, we evaluated the definite integral $\int_1^2 x e^x \, dx$ using the technique of integration by parts. We showed that the integral can be evaluated by choosing $u = x$ and $v = e^x$ , and then using the product rule of differentiation to find the derivative of the product $uv$ . We then integrated both sides of the equation with respect to $x$ and evaluated the resulting expression to find the value of the integral.

References

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

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = x * sp.exp(x)

# Evaluate the integral
result = sp.integrate(f, (x, 1, 2))

print(result)

This code uses the Sympy library to evaluate the integral $\int_1^2 x e^x \, dx$ . The result is printed to the console.

Introduction

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

Q1: What is the concept of integration by parts?

A1: Integration by parts is a technique used to evaluate integrals of the form $\int u \, dv$ . It is based on the product rule of differentiation, which states that if $u$ and $v$ are two functions of $x$ , then the derivative of their product is given by:

\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}

Q2: How do I choose the functions $u$ and $v$ for integration by parts?

A2: To choose the functions $u$ and $v$ , you need to identify the two functions that are being multiplied together in the integral. Then, you can choose one of the functions to be $u$ and the other to be $v$ . For example, in the integral $\int x e^x \, dx$ , we can choose $u = x$ and $v = e^x$ .

Q3: What is the formula for integration by parts?

A3: The formula for integration by parts is:

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

Q4: How do I evaluate the integral $\int_1^2 x e^x \, dx$ using integration by parts?

A4: To evaluate the integral $\int_1^2 x e^x \, dx$ using integration by parts, we can choose $u = x$ and $v = e^x$ . Then, we have:

\frac{du}{dx} = 1 \quad \text{and} \quad \frac{dv}{dx} = e^x

Substituting these values into the formula, we get:

\int x e^x \, dx = x e^x - \int e^x \, dx

Q5: What is the value of the integral $\int_1^2 x e^x \, dx$ ?

A5: The value of the integral $\int_1^2 x e^x \, dx$ is:

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

Q6: Can I use other techniques to evaluate the integral $\int_1^2 x e^x \, dx$ ?

A6: Yes, you can use other techniques to evaluate the integral $\int_1^2 x e^x \, dx$ . For example, you can use the substitution method or the integration by substitution method.

Q7: What is the substitution method?

A7: The substitution method is a technique used to evaluate integrals of the form $\int f(g(x)) \, dx$ . It involves substituting $g(x)$ with a new variable, say $u$ , and then integrating with respect to $u$ .

Q8: Can I use the substitution method to evaluate the integral $\int_1^2 x e^x \, dx$ ?

A8: Yes, you can use the substitution method to evaluate the integral $\int_1^2 x e^x \, dx$ . For example, you can substitute $u = e^x$ and then integrate with respect to $u$ .

Q9: What is the value of the integral $\int_1^2 x e^x \, dx$ using the substitution method?

A9: The value of the integral $\int_1^2 x e^x \, dx$ using the substitution method is:

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

Q10: Can I use a calculator or computer software to evaluate the integral $\int_1^2 x e^x \, dx$ ?

A10: Yes, you can use a calculator or computer software to evaluate the integral $\int_1^2 x e^x \, dx$ . For example, you can use the Sympy library in Python to evaluate the integral.

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = x * sp.exp(x)

# Evaluate the integral
result = sp.integrate(f, (x, 1, 2))

print(result)

This code uses the Sympy library to evaluate the integral $\int_1^2 x e^x \, dx$ . The result is printed to the console.