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Magazine Chapter 5: Problem 2
Write the general antiderivative. \(\int 2 x e^{x^{2}} d x\)
Short Answer
Expert verified
The general antiderivative is \(e^{x^2} + C\).
Step by step solution
01
Recognize the Form of the Integral
The given integral is \(\int 2x e^{x^2} dx\). This suggests using a substitution method because it resembles the form \(\int u' e^u du\).
02
Choose the Substitution
Let \(u = x^2\). Thus, the differential \(du = 2x \, dx\), which exactly matches the integrand's derivative. This simplifies and allows direct substitution.
03
Substitute into the Integral
Substitute \(u = x^2\) and \(du = 2x \, dx\) into the integral, transforming it to \(\int e^u \, du\). This is a basic exponential integral.
04
Integrate with Respect to u
The integral of \(e^u\) with respect to \(u\) is \(e^u + C\), where \(C\) is the constant of integration.
05
Replace the Substitution to Original Variable
Substitute back \(x^2\) for \(u\) to express the solution in terms of \(x\). Therefore, the general antiderivative is \(e^{x^2} + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Antiderivative
An antiderivative, also known as an indefinite integral, is a fundamental concept in calculus. It refers to a function whose derivative is the given function. When you take the antiderivative, you essentially reverse the process of differentiation.
Key points to remember are:
Recognizing the form of a function is crucial in finding its antiderivative. Identifying patterns, such as when a function resembles a known derivative or fits a specific technique like substitution, helps solve the integral efficiently.
Key points to remember are:
- The result of an antiderivative includes a "+ C" term. Since differentiation of a constant is zero, any constant could have been added, hence the need for "+ C".
- Not every function has a simple antiderivative that can be easily written using elementary functions. Sometimes, techniques like substitution are required to find it.
- The notation for the antiderivative is \( int f(x)\,dx\).
Recognizing the form of a function is crucial in finding its antiderivative. Identifying patterns, such as when a function resembles a known derivative or fits a specific technique like substitution, helps solve the integral efficiently.
Substitution Method
The substitution method is a powerful tool in calculus, often used to simplify complex integrals. It's akin to a change of variables, where you substitute one variable for another to make the integration process easier.
Here's how it works:
The key advantage of substitution is that it transforms a potentially tricky integral into a form that is much more straightforward to solve. In the example of \( int 2x e^{x^2} \, dx \), by letting \(u = x^2\), the integral readily becomes \( int e^u \, du \,\) which is much easier to integrate.
Here's how it works:
- You start by identifying a part of the integrand (the expression inside the integral) as a new variable. This helps simplify the integral's structure.
- Next, find the derivative of this new variable, which corresponds to a differential necessary for the substitution.
- Replace the identified parts of your integral with the new variable and its derivative.
- Integrate the simplified expression.
- Finally, substitute back the original variable into the solution.
The key advantage of substitution is that it transforms a potentially tricky integral into a form that is much more straightforward to solve. In the example of \( int 2x e^{x^2} \, dx \), by letting \(u = x^2\), the integral readily becomes \( int e^u \, du \,\) which is much easier to integrate.
Exponential Functions
Exponential functions are functions of the form \(f(x) = a^x\), where the base \(a\) is a constant. When the base is the natural number \(e\), approximately 2.718, the function is called the natural exponential function and is denoted as \(e^x\).
Some important features include:
In integrals, like \( int e^{u} \, du\), finding the antiderivative is straightforward because the function retains its form. Thus, the integral is simply \(e^u + C\). This simplicity often arises due to using substitution to reduce more complex expressions into basic exponential forms.
Some important features include:
- The derivative of \(e^x\) is \(e^x\), which makes it unique and particularly interesting for solving differential equations.
- The integral of \(e^x\) is also \(e^x\), up to a constant. This is why exponential functions often appear in problems involving growth or decay in fields like biology, finance, and physics.
- Exponential functions grow or decay at rates proportional to their current value, leading to the exponential growth or decay process.
In integrals, like \( int e^{u} \, du\), finding the antiderivative is straightforward because the function retains its form. Thus, the integral is simply \(e^u + C\). This simplicity often arises due to using substitution to reduce more complex expressions into basic exponential forms.
