Problem 3 - Applied Calculus

Question

Find an antiderivative. $$ g(t)=t^{2}+t $$

Step-by-Step Solution

Answer

An antiderivative is \( G(t) = \frac{t^3}{3} + \frac{t^2}{2} + C \).

Step 1: Identify the function to integrate

We are given the function \( g(t) = t^2 + t \). Our goal is to find an antiderivative of this function, which means we need to find a function \( G(t) \) whose derivative is \( g(t) \).

Step 2: Apply the power rule of integration

The power rule for integration states that \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), where \( C \) is a constant of integration. We will apply this rule to each term of \( g(t) = t^2 + t \).

Step 3: Integrate each term individually

Start by integrating the first term: \( \int t^2 \, dt = \frac{t^{2+1}}{2+1} = \frac{t^3}{3} \). Next, integrate the second term: \( \int t \, dt = \frac{t^{1+1}}{1+1} = \frac{t^2}{2} \).

Step 4: Combine the results and add the constant of integration

Combine the results from the previous step: \( G(t) = \frac{t^3}{3} + \frac{t^2}{2} + C \). Ensure you add \( C \), the constant of integration, to account for all possible antiderivatives.

Key Concepts

Integration

Integration is a fundamental concept in calculus, acting as the reverse operation of differentiation. While differentiation finds the rate of change or the slope of a function, integration finds the total accumulation, essentially gathering parts to form a whole.

In more simple terms, if differentiation involves cutting something apart to see what makes it work, integration puts those pieces back together to form the original whole. When we talk about integrating a function, we are genuinely interested in discovering what function, when differentiated, gets us back to our original problem.

To integrate a function, we apply rules and formulas in a determined sequence. Among these, the power rule, one of the most common rules, serves as a friendly method for dealing with polynomials and functions that can be expressed as powers of variables. More details on this will follow in the next section!

Power Rule

When dealing with integration, the power rule is a straightforward yet powerful tool in our mathematical toolbox. It applies to functions of the form \(t^n\), where \(n\) is any real number except \(-1\).

To use the power rule, simply follow a couple of easy steps:

• Add 1 to the exponent \(n\).
• Divide the new expression by this new exponent.

Mathematically, the power rule for integration is expressed as:
\[ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \]
This formula tells us to add one to the power and divide by the new power. The \(C\) represents the constant of integration, which we'll explain further in the next section. Just remember, applying the power rule is as easy as follow-the-steps, where simple arithmetic leads you to your answer!

Constant of Integration

The constant of integration, denoted as \(C\), is a unique aspect of indefinite integration. When finding the antiderivative of a function, it's important to include this constant.

This necessity arises because the differentiation of a constant (just a plain number) yields zero. Therefore, when integrating and seeking the original function, we must account for all possible constants that could have differentiated to become part of the result.

• Think of this constant as a placeholder representing all those possibilities.
• It ensures our antiderivative encompasses every potential solution.

An example helps: when integrating \(t^2 + t\), the result \(\frac{t^3}{3} + \frac{t^2}{2} + C\) should always include this \(C\). Always remember, every antiderivative gets a little more flexible thanks to the constant of integration!