Question

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(x^{3}+3 x+9\right)\left(x^{2}+1\right) d x $$

Short answer

The indefinite integral of \((e^{x}-2)^{2}\) is \(\frac{1}{2}*e^{2x} - 4e^{x} + 4x + C\).

Step-by-step solution

Expand the Square

Start by expanding the square in the integral. This gives \((e^{x})^2 - 2*2*e^{x} + 2^2\), which simplifies to \(e^{2x} - 4e^{x} + 4\).

Simplify the Integral

Now rewrite the integral as follows: \(\int e^{2x} - 4e^{x} + 4 dx\).

Perform the Integration

The integral of \(e^{2x}\) will require a substitution. Let \(u = 2x\), then \(du = 2dx\) or \(dx = du/2\). For each term in the integral, the result is as follows: \(\int e^{u} *(du/2) - 4\int e^{x} dx + 4\int dx\). This simplifies to \(\frac{1}{2}\int e^{u} du - 4\int e^{x} dx + 4\int dx\).

Calculate the Individual Integrals

Now calculate the integrals: \(\frac{1}{2}*e^{u} - 4e^{x} + 4x + C\).

Replace the Substituted Variable

Finally, replace \(u\) with the original \(2x\). The result is \(\frac{1}{2}*e^{2x} - 4e^{x} + 4x + C\).

Key concepts

Symbolic Integration

Symbolic integration is a fundamental concept in calculus, which involves finding the antiderivative, or the indefinite integral, of an algebraic expression. It requires the application of integration rules and methods to evaluate the integral symbolically, without necessarily resorting to numerical methods. For instance, the symbolic integration of a quadratic expression like \( (e^x - 2)^2 \) requires expanding and simplifying the square to obtain the terms that can be integrated term by term. This process not only simplifies the computation, but also leads to more insight into the integrand's behavior.

Symbolic integrators, often built into computer algebra systems, assist in carrying out integration symbolically. These tools are especially valuable when dealing with complex integrals that are difficult to solve manually. They apply algorithms based on the properties of integrals and known antiderivatives to yield results that are accurate and in expressible algebraic form.

Integration By Substitution

Integration by substitution, also known as u-substitution, is a technique that simplifies the integration process by changing the variable of integration to make the integral more manageable. This method is particularly useful when the integrand is a composition of functions that can be disentangled by a substitution.

For example, in the integral \( \int e^{2x} dx \), the substitution \( u = 2x \) transforms the integrand into \( e^u \) with \( du = 2dx \) or \( dx = \frac{du}{2} \). This converts the original integral into a simpler form that leverages the basic antiderivative of the exponential function. After integration, the original variable is substituted back in to achieve the final result. Understanding this method is crucial for tackling a wide range of integrals and is a staple in calculus courses.

Expanding the Square

Expanding the square is an algebraic technique used to simplify expressions that include a squared binomial. The process of expansion follows the pattern \( (a - b)^2 = a^2 - 2ab + b^2 \), helping to break down complex expressions into simpler terms that can be easily handled in calculus operations like integration.

In the given exercise, expanding the square of \( (e^x - 2)^2 \) simplifies to \( e^{2x} - 4e^x + 4 \). By doing so, we've transformed a challenging integral into a sum of simpler terms, each of which can be integrated separately. This approach not only streamlines the integration process but also aligns with the linearity property of integrals, which states that the integral of a sum is the sum of the integrals.

Antiderivatives

Antiderivatives play a pivotal role in calculus, representing the reverse process of taking derivatives. When calculating an indefinite integral, one is actually seeking the antiderivative of the integrand. The antiderivative is a more general function that, when differentiated, yields the original expression. For example, the antiderivative of \( e^x \) is itself \( e^x \) plus a constant \( C \), accounting for the family of possible functions that share the same derivative.

It is important to understand the various rules and techniques for finding antiderivatives, such as the power rule, integration by parts, and trigonometric integrals, allowing one to handle a wide array of integrals. The process of finding antiderivatives is not always straightforward, but with practice and a solid grasp of derivative rules, it becomes an accessible and logical task.

Exponential Functions Integration

The integration of exponential functions involves finding the antiderivative of expressions in the form of \( a^x \), where \( a \) is a constant. Exponential functions have the elegant property that their derivative is proportional to the original function. Specifically, the integral of \( e^x \) is \( e^x \) plus a constant \( C \).

This property simplifies the integration process when dealing with expressions that include exponential terms. For more involved exponential integrals, such as \( \int e^{2x} dx \), substitution or other methods might be necessary to align the integrand with the basic form \( \int e^x dx \). Understanding how to work with exponential integrals is crucial in advanced mathematics, physics, and engineering, as exponential growth patterns are widespread in natural and social phenomena.