Question

Find each indefinite integral. Check some by calculator. $$\int\left(9-x^{2}\right) d x$$

Short answer

\[\int(9 - x^2)dx = 9x - \frac{x^3}{3} + C\]

Step-by-step solution

Apply the Linearity of the Integral

The integral of a sum of two functions is the sum of the integrals of the individual functions. So, we split the given integral \(\int(9 - x^2)dx\) into \(\int 9 dx - \int x^2 dx\).

Integrate the Constant Term

The integral of a constant \(a\) with respect to \(x\) is \(ax\), so for \(\int 9 dx\), we have \(9x\).

Integrate the Variable Term

The integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\) for any real number \(n \eq -1\). Using this formula, \(\int x^2 dx\) becomes \(\frac{x^{3}}{3}\).

Combine the Integrals

Combining the results from Step 2 and Step 3, and adding the constant of integration \(C\), the indefinite integral is \(9x - \frac{x^{3}}{3} + C\).

Key concepts

Understanding Integration Techniques

Integration techniques are fundamental tools in calculus, involving several methods to find the indefinite integral or antiderivative of a function. When you come across an integral, such as \(\int(9 - x^2)dx\), your first step should be to identify which technique best applies. In this example, we use one of the simplest techniques, the linearity rule, which states that the integral of a sum (or difference) of functions is equal to the sum (or difference) of their integrals.

What's essential here is to recognize how to handle different types of terms—constant vs. variable. In our case, \(9\) is a constant and \(x^2\) is a variable term. Each of these is integrated separately using basic formulas and rules for integration. After obtaining the integrals individually, we combine them to get the final solution. It can be beneficial to verify your result with a calculator or integration software to ensure accuracy.

Exploring the Linearity of Integrals

The linearity of integrals is another critical concept to grasp as it underpins many integration problems you will encounter. Simply put, it allows for the breaking down of complex integrals into simpler parts that are more manageable to solve. The problem \(\int(9 - x^2)dx\) is a perfect illustration of this principle in action. Here, we split the integral into two parts: \(\int 9 dx\) and \(\int x^2 dx\) and proceed to integrate each part separately.

Step-by-Step Application

Step 1 involves leveraging linearity to separate the constant term from the variable term. Steps 2 and 3 focus on integrating these terms independently. The constant integrates to \(9x\) and for the variable term, the power rule for integration is applied to obtain \(\frac{x^3}{3}\). Remember that a constant of integration, typically denoted by \(C\), is always added to the result of an indefinite integral, as it represents the family of all its antiderivatives.

The Role of Antiderivatives in Integration

An antiderivative of a function is another function whose derivative is the original function. In the context of integration, finding an antiderivative equates to solving an indefinite integral, like the example \(\int(9 - x^2)dx\).

The concept of antiderivatives is crucial because it provides a general method to reverse the process of differentiation. When you integrate \(9\), you're looking for a function which, when differentiated, gives you \(9\), and the answer is \(9x\) as derivatives of constants are zero. Similarly, integrating \(x^2\) means finding a function which gives \(x^2\) when differentiated, leading us to \(\frac{x^3}{3}\) using the power rule of differentiation. The antiderivative is not unique due to the arbitrary constant \(C\), thus encompassing an infinite set of functions that differentiate to the original function.