Chapter 1: Problem 31
For the following exercises, find the antiderivatives for the given functions.\(x \cosh \left(x^{2}\right)\)
Short Answer
Expert verified
The antiderivative is \(\frac{1}{2} \sinh(x^2) + C\).
Step by step solution
01
Identify the Function and Method
The function given is \(x \cosh(x^2)\). We need to find its antiderivative. The appropriate method for finding this antiderivative is through substitution, as the presence of \(x^2\) suggests a natural substitution.
02
Choose the Substitution
Let \(u = x^2\). Then, the differential is \(du = 2x \, dx\). Solving for \(x \, dx\), we get \(x \, dx = \frac{1}{2} du\). We will substitute these expressions into the integral.
03
Substitute and Simplify
Substitute \(u = x^2\) and \(x \, dx = \frac{1}{2} du\) into the integral: \( \int x \cosh(x^2) \, dx = \int \cosh(u) \frac{1}{2} du \). Simplifying gives \( \frac{1}{2} \int \cosh(u) \, du \).
04
Integrate
The integral of \(\cosh(u)\) is \(\sinh(u)\). Thus, \(\frac{1}{2} \int \cosh(u) \, du = \frac{1}{2} \sinh(u) + C\), where \(C\) is the constant of integration.
05
Back-Substitute
Replace \(u\) with \(x^2\) to express the answer in terms of the original variable: \(\frac{1}{2} \sinh(x^2) + C\). Thus, the antiderivative of \(x \cosh(x^2)\) is \(\frac{1}{2} \sinh(x^2) + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method in integration is a powerful technique that simplifies the process of finding antiderivatives, especially for composite functions. When you look at the function \(x \cosh(x^2)\), the presence of \(x^2\) signals a chance to use this method. The idea is to replace a part of the integrand with a new variable \(u\), making the integral easier to handle.
To utilize the substitution method, we choose \(u = x^2\). This choice efficiently transforms the problem because the differential of \(u\), which is \(du = 2x \, dx\), matches well with the \(x \, dx\) in our integrand. By doing this substitution, we're transitioning from an integral in terms of \(x\) to one in terms of \(u\), which often simplifies the computation. In this case, you substitute \(x \, dx = \frac{1}{2} \, du\), and the integral becomes \(\frac{1}{2} \int \cosh(u) \, du\).
This technique fundamentally rests on recognizing parts of the integrand that could be transformed into a simpler form using a strategic substitution.
To utilize the substitution method, we choose \(u = x^2\). This choice efficiently transforms the problem because the differential of \(u\), which is \(du = 2x \, dx\), matches well with the \(x \, dx\) in our integrand. By doing this substitution, we're transitioning from an integral in terms of \(x\) to one in terms of \(u\), which often simplifies the computation. In this case, you substitute \(x \, dx = \frac{1}{2} \, du\), and the integral becomes \(\frac{1}{2} \int \cosh(u) \, du\).
This technique fundamentally rests on recognizing parts of the integrand that could be transformed into a simpler form using a strategic substitution.
Hyperbolic Functions
Hyperbolic functions, like their trigonometric siblings, arise from the exponential function. They are particularly useful in various fields, such as calculus and physics. The function \(\cosh(x)\), or the hyperbolic cosine, is one such function and is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
In calculus, hyperbolic functions often appear in integration problems due to their appealing properties and simple derivatives. For instance, the derivative of \(\sinh(x)\) is \(\cosh(x)\), making these functions integrally linked. In our problem, once the substitution \(u = x^2\) is applied, we're left to integrate the hyperbolic cosine, \(\cosh(u)\). The integration process capitalizes on the direct relationship between \(\sinh(x)\) and \(\cosh(x)\).
Knowing the antiderivative of \(\cosh(u)\) is \(\sinh(u)\) is critical, as it allows us to easily solve the integral as \(\frac{1}{2}\sinh(u) + C\). This showcases why hyperbolic functions are favored in integration -- they simplify the process when appearing in substitution or through direct computation.
In calculus, hyperbolic functions often appear in integration problems due to their appealing properties and simple derivatives. For instance, the derivative of \(\sinh(x)\) is \(\cosh(x)\), making these functions integrally linked. In our problem, once the substitution \(u = x^2\) is applied, we're left to integrate the hyperbolic cosine, \(\cosh(u)\). The integration process capitalizes on the direct relationship between \(\sinh(x)\) and \(\cosh(x)\).
Knowing the antiderivative of \(\cosh(u)\) is \(\sinh(u)\) is critical, as it allows us to easily solve the integral as \(\frac{1}{2}\sinh(u) + C\). This showcases why hyperbolic functions are favored in integration -- they simplify the process when appearing in substitution or through direct computation.
Integration Techniques
Integration techniques encompass various methods to tackle different types of integrals, and choosing the right one often determines the ease of solving the problem. Common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric identities.
The choice of technique hinges on recognizing patterns and structures within the function. Substitution, as applied here, is particularly useful for functions involving compositions, like \(x \cosh(x^2)\). It can replace more complicated expressions with simpler ones, facilitating integration.
Besides substitution, being familiar with basic integral forms, such as those involving hyperbolic functions, is a vital technique. Knowing that \(\int \cosh(u) \, du = \sinh(u) + C\) allows us to integrate efficiently without complicated steps. This knowledge reduces the cognitive load and often provides solutions with less room for error.
Combining the understanding of when to use substitution alongside recognizing integral forms streamlines the integration process, enhancing problem-solving efficiency. Tackling every integral problem starts with this foundational decision: what technique applies best, and leveraging it appropriately becomes the key to mastering integration.
The choice of technique hinges on recognizing patterns and structures within the function. Substitution, as applied here, is particularly useful for functions involving compositions, like \(x \cosh(x^2)\). It can replace more complicated expressions with simpler ones, facilitating integration.
Besides substitution, being familiar with basic integral forms, such as those involving hyperbolic functions, is a vital technique. Knowing that \(\int \cosh(u) \, du = \sinh(u) + C\) allows us to integrate efficiently without complicated steps. This knowledge reduces the cognitive load and often provides solutions with less room for error.
Combining the understanding of when to use substitution alongside recognizing integral forms streamlines the integration process, enhancing problem-solving efficiency. Tackling every integral problem starts with this foundational decision: what technique applies best, and leveraging it appropriately becomes the key to mastering integration.
