Question

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(t^{2}-\frac{1}{t^{2}}\right) d t$$

Short answer

\((1/3)t^3 - t^{-1} + C\)

Step-by-step solution

Integration of the first term

The first term of the integral is \(t^2\). Using the power rule that the integral of \(x^n\) with respect to \(x\) is \(1/(n+1)x^{n+1}\), this term integrates to \((1/3)t^3\).

Integration of the second term

The second term of the integral is \(-1/t^2\). This term can be rewritten as \(-t^{-2}\). Now, we apply the power rule again and this term integrates to \(t^{-1}\).

Combining the results

The entire integral is thus \((1/3)t^3 - t^{-1}\) plus the constant of integration, which we will represent by \(C\). So the final answer is \((1/3)t^3 - t^{-1} + C\).

Key concepts

Integration Techniques

Integration is a fundamental concept in calculus, used to find the area under a curve, among other things. There are various techniques to tackle different integrals, each suited to a specific type of function or equation. For instance, in the exercise \( \int(t^{2}-\frac{1}{t^{2}}) dt \), the function consists of two terms, so we can integrate each term separately—a technique known as term-by-term integration.

Other techniques include substitution, where one part of the integral is replaced with a single variable to simplify the calculation, and integration by parts, which is useful for products of functions. Nonetheless, no matter the technique, all indefinite integrals will include a constant of integration to account for the fact that the antiderivative is not unique. Understanding when and how to use these techniques is crucial for solving integration problems effectively.

Power Rule for Integration

The power rule for integration is a basic yet powerful tool. It provides a quick method to integrate functions of the form \( x^n \) with respect to \( x \). The rule states that the integral of \( x^n \) is \( \frac{1}{n+1}x^{n+1} \), provided that \( n eq -1 \).

In our exercise, we apply the power rule twice: once to the term \( t^2 \) which becomes \( \frac{1}{3}t^3 \) after applying the power rule, and the second time to the term \(-\frac{1}{t^{2}}\), which we first rewrite as \( -t^{-2} \) to fit the rule. This term integrates to \( -t^{-1} \) or \( -\frac{1}{t} \). It's important to remember that the power rule for integration simplifies the process greatly and is an essential part of a student's integration toolkit.

Constant of Integration

The constant of integration, often denoted as \( C \), is a critical part of any indefinite integral. Because the derivative of a constant is zero, when we find the antiderivative of a function, there are infinitely many solutions which differ by a constant amount. Therefore, we include \( C \) to represent this family of solutions.

In the solution to the given exercise, after applying the power rule for integration to both terms \( \frac{1}{3}t^3 \) and \( -t^{-1} \) respectively, we combine them and add the constant of integration to get the final answer \( \frac{1}{3}t^3 - \frac{1}{t} + C \.\) Understanding the importance of \( C \) allows students to comprehend that any indefinite integral represents an infinite set of possible functions, all varying by just a constant.