Problem 10 - Calculus and its applications

Question

Find each integral. $$ \int\left(3 t^{2}-4 t+7\right) d t $$

Step-by-Step Solution

Answer

The integral is \( t^3 - 2t^2 + 7t + C \).

Step 1: Understand the Problem

We are given a polynomial function inside an integral: \( \int (3t^2 - 4t + 7) \, dt \). Our task is to find the indefinite integral, also known as the antiderivative, of this expression with respect to \( t \).

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int t^n \, dt = \frac{1}{n+1} t^{n+1} + C \) for \( n eq -1 \). We will apply this rule to each term of the polynomial separately.

Step 3: Integrate Each Term Separately

1. For \( 3t^2 \), integrate to get \( \int 3t^2 \, dt = 3 \cdot \frac{1}{3} t^{2+1} = t^3 \).2. For \( -4t \), integrate to get \( \int -4t \, dt = -4 \cdot \frac{1}{2} t^{1+1} = -2t^2 \).3. For the constant \( 7 \), integrate to get \( \int 7 \, dt = 7t \).

Step 4: Combine the Integrated Terms

Add the results from Step 3 to get the integrated function: \( t^3 - 2t^2 + 7t + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule

The power rule is a vital tool in calculus when dealing with integration and differentiation. In the context of integration, the power rule specifically helps us to integrate expressions that have variable terms raised to a power. When you see something like \( t^n \) in an integral, that's where the power rule shines. The rule can be summarized by the formula: \( \int t^n \, dt = \frac{1}{n+1} t^{n+1} + C \) for \( n eq -1 \). This formula allows us to "raise the power" of the term and divide by the new exponent.

Here's how you apply it:

• Identify the term with the power, which is usually the variable raised to some degree.
• Increase the exponent by one.
• Divide the coefficient of this term by the new exponent.
• Don’t forget to add the constant of integration \( C \), as this accounts for any constant that might have been part of the original function before differentiation.

For example, when integrating the term \( 3t^2 \), you apply the power rule and end up with \( t^3 \), since \( 3 \times \frac{1}{3}t^{3} = t^3 \). This makes solving integrals much smoother and is essential for finding antiderivatives in polynomial expressions.

Indefinite Integral

The term "indefinite integral" refers to the process of finding the antiderivative of a function. Unlike definite integrals, indefinite integrals do not have set limits. Thus, they provide us with a family of functions that share a common derivative.

When you integrate a function without specified upper and lower bounds, you end up with an indefinite integral. This form always includes a constant of integration, \( C \), because any constant difference in the original function disappears upon differentiation.

Understanding indefinite integrals is essential in calculus as they allow us to deduce the original function given its derivative. They're often represented with the integral sign \( \int \) followed by a function and a differential, like \( \int (3t^2 - 4t + 7) \, dt \).

This process essentially "undoes" differentiation, allowing us to explore the relationships between functions and their rates of change. The solution to our example problem demonstrates this, resulting in \( t^3 - 2t^2 + 7t + C \).

Antiderivative

Finding the antiderivative of a function is akin to retracing steps in calculus to determine the original function from its derivative. The antiderivative is essentially the function that had undergone differentiation to give the original function we're given. There is often more than one antiderivative, thanks to the constant \( C \).

In this specific problem, we are tasked with finding the antiderivative of the polynomial \( 3t^2 - 4t + 7 \). By integrating each term separately:

• Integrate \( 3t^2 \) and get \( t^3 \).
• Integrate \( -4t \) and get \( -2t^2 \).
• Integrate the constant \( 7 \) to obtain \( 7t \).

Combining these results with our understanding of constants, we arrive at the antiderivative \( t^3 - 2t^2 + 7t + C \).

This concept of antiderivatives is crucial for solving real-world problems where it's important to revert from a rate of change to the original quantity or position.