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Magazine Chapter 7: Problem 19
Find the antiderivatives. \(\int x e^{x^{2}} d x\)
Short Answer
Expert verified
The antiderivative is \( \frac{1}{2}e^{x^2} + C \).
Step by step solution
01
Recognize the Integration Method
The integral \( \int x e^{x^2} \, dx \) involves a function and its derivative, which suggests using the technique of substitution.
02
Choose a Substitution
Set \( u = x^2 \). Then, the derivative \( du = 2x \, dx \) implies that \( x \, dx = \frac{1}{2} du \). This substitution will simplify the integration process.
03
Rewrite the Integral
Substitute \( u = x^2 \) and \( x \, dx = \frac{1}{2} du \) into the integral: \[ \int x e^{x^2} \, dx = \int e^u \cdot \frac{1}{2} du = \frac{1}{2} \int e^u \, du \]
04
Integrate with Respect to New Variable
The integral \( \int e^u \, du \) is straightforward and equal to \( e^u + C \). Therefore, we have: \[ \frac{1}{2} \int e^u \, du = \frac{1}{2}(e^u) + C = \frac{1}{2}e^u + C \]
05
Substitute Back to Original Variable
Replace \( u \) with \( x^2 \) to express the antiderivative in terms of the original variable: \[ \frac{1}{2}e^u + C = \frac{1}{2}e^{x^2} + C \]
06
Final Antiderivative
The final expression for the antiderivative of \( \int x e^{x^2} \, dx \) is: \[ \frac{1}{2}e^{x^2} + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a crucial technique in calculus, especially for finding antiderivatives of complex functions. It helps simplify an integral by transforming it into a simpler form. This method is akin to reversing the chain rule from differentiation. Let's break down the process:
- Start by identifying a part of the integrand, typically the "inner function," that would be suitable for substitution.
- Choose a new variable, say \( u \). For example, if you have an expression like \( x^2 \), you might let \( u = x^2 \).
- Next, compute the differential \( du \) by finding the derivative of \( u \) with respect to \( x \). In our case, \( du = 2x \, dx \).
- The goal is to express the original integral entirely in terms of \( u \) and \( du \). This often involves rewriting \( dx \) or other terms as fractions or multiples of \( du \).
- Once the integral is rewritten, integrate with respect to \( u \).
- Finally, substitute back the original variable to express the solution in terms of the initial variable from the problem.
Integrals
Integrals are a fundamental concept in calculus, serving as the reverse process of differentiation. They allow for the computation of areas under curves, among other applications. Here's a clearer understanding:
- Indefinite integrals represent a family of functions and are usually written without bounds. They provide an antiderivative of the function being integrated.
- Definite integrals calculate the actual area under a curve over a specified interval. They have upper and lower limits.
- The process of integration adds up infinitesimally small quantities to determine a whole, which can be expressed mathematically through the integral sign \( \int \).
- The result of an indefinite integral always includes an arbitrary constant \( C \), reflecting the fact that antiderivatives are determined up to a constant. This accounts for all possible vertical shifts of the antiderivative.
Exponential Functions
Exponential functions are potent mathematical functions where the variable is in the exponent. A common exponential function is \( e^x \), which is the base of natural logarithms. These functions have unique characteristics:
- The derivative and the integral of the exponential function \( e^x \) are both \( e^x \), showcasing a unique property among functions.
- Exponential functions grow rapidly, making them significant in modeling phenomena like population growth and radioactive decay.
- When integrating exponential functions, properties like constant multiplication can simplify the process. For instance, the integral \( \int e^u \, du = e^u + C \) is direct and straightforward.
- Understanding exponential functions in integration, especially through substitution, is key. Substitution often involves transforming an exponent into a new variable, as in the exercise where \( x^2 \) becomes \( u \).
