Problem 30 - Applied Calculus

Question

Find an antiderivative. $$r(t)=\frac{1}{t^{2}}$$

Step-by-Step Solution

Answer

The antiderivative of \( \frac{1}{t^2} \) is \( -\frac{1}{t} + C \).

Step 1: Identify the Function Type

The given function is \( r(t) = \frac{1}{t^2} \). We need to find an antiderivative, which involves integrating this function. Recognize that \( \frac{1}{t^2} \) can be rewritten as \( t^{-2} \).

Step 2: Apply the Power Rule for Integration

To integrate \( t^{-2} \), we use the power rule for integration. The power rule states that \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), where \( n eq -1 \). For \( t^{-2} \), we have \( n = -2 \).

Step 3: Compute the Antiderivative

Using the power rule, integrate \( t^{-2} \, dt \). Substitute \( n=-2 \) into the formula: \( \int t^{-2} \, dt = \frac{t^{-2+1}}{-2+1} + C = \frac{t^{-1}}{-1} + C = -t^{-1} + C \).

Step 4: Rewrite the Expression

Since \( -t^{-1} \) can also be expressed as \( -\frac{1}{t} \), the antiderivative of \( \frac{1}{t^2} \) is \( -\frac{1}{t} + C \).

Key Concepts

Antiderivative

When we talk about finding an antiderivative, we are essentially discussing the process of integration, which is the reverse operation of differentiation. An antiderivative of a function is a new function whose derivative is the original function. For example, if you differentiate the function \(F(x)\), you will recover the function \(f(x)\) that you started with. The solution provided begins with the goal of finding an antiderivative for \(r(t) = \frac{1}{t^2}\). The general solution will have a constant, expressed as \(C\), because when differentiating, any constant has a derivative of zero, meaning it doesn’t affect the original function.

• An antiderivative can be thought of as an indefinite integral, represented as \(\int f(x) \, dx\).
• The result of this integration always includes a \(+ C\), denoting the constant of integration.
• This is because the process of finding an antiderivative is akin to "undoing" differentiation.

Power Rule

The power rule is one of the simplest and most useful rules for finding antiderivatives. It helps in integrating power functions. In our case, the given function \(r(t) = \frac{1}{t^2}\) can be transformed into a power form by rewriting it as \(t^{-2}\). This makes it easier to apply the power rule, which is defined as \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\), given \(n eq -1\).

• The power rule allows us to integrate functions of the form \(t^n\) where \(n eq -1\).
• By increasing the power of \(t\) by one and dividing by the new exponent, we find the antiderivative.
• This rule simplifies integration for a wide variety of polynomial expressions.

After determining that \(n\) is \(-2\) for our example, applying the power rule yields \(\int t^{-2} \, dt = \frac{t^{-1}}{-1} + C\), simplifying to \(-t^{-1} + C\). This transforms back to \(-\frac{1}{t} + C\), which is the required antiderivative.

Integration Techniques

When encountering a function to integrate, various techniques are available to find the antiderivative efficiently. Simple functions like polynomials often require just the application of the power rule, but more complex functions might need other strategies.For \(r(t) = \frac{1}{t^2} = t^{-2}\), using the power rule is straightforward. However, if you had a function with multiple terms or required multiple rules, other techniques would be necessary:

• Substitution: Useful when a function involves composite functions, making the integration process easier by substituting part of the expression with a single variable.
• Integration by Parts: Applicable when the integrand is a product of two functions. It relies on the differentiation counterpart: the product rule.
• Partial Fraction Decomposition: Useful for rational functions, by breaking them into simpler fractions which can often be integrated individually.

For this specific exercise, directly applying the power rule for the integration of a simple function like \(t^{-2}\) is adequate, highlighting the rule's efficiency for single-term power functions.