Problem 19 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

Step-by-Step Solution

Answer

The most general antiderivative is \( t^3 + \frac{t^2}{4} + C \).

Step 1: Identify Separate Terms

The integrand consists of two terms: \(3t^2\) and \(\frac{t}{2}\). We can integrate each term separately according to the properties of indefinite integrals.

Step 2: Integrate the First Term

Apply the power rule for integration to the first term: \( \int 3t^2 \, dt \). According to the power rule, \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \). Here, \( n = 2 \), so we have \( \frac{3t^{3}}{3} = t^3 \).

Step 3: Integrate the Second Term

For the second term, apply the power rule again: \( \int \frac{t}{2} \, dt \). Treat \( \frac{1}{2} \) as a constant, giving us \( \frac{1}{2} \int t \, dt \). By the power rule, \( \frac{1}{2} \cdot \frac{t^2}{2} = \frac{t^2}{4} \).

Step 4: Combine the Results

Combine the antiderivatives found in Steps 2 and 3: \( t^3 + \frac{t^2}{4} + C \), where \( C \) is the constant of integration.

Step 5: Verify by Differentiation

Differentiate the result \( t^3 + \frac{t^2}{4} + C \) to verify correctness. The derivative is \( 3t^2 + \frac{t}{2} \), which matches our original integrand. Therefore, the integration is correct.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus. It's a method used to find the antiderivative of polynomial expressions. The rule states:

• If you have an expression in the form of \( t^n \), its integral is \( \frac{t^{n+1}}{n+1} \), where \( n eq -1 \).
• Always add a constant of integration, \( C \), at the end.

This rule transforms an algebraic expression into its antiderivative by increasing the exponent by one and dividing by the new exponent.
For example, to integrate \( 3t^2 \), we apply the power rule:

• Increase the exponent of \( t^2 \) to 3, leading to \( t^3 \).
• Then divide by the new exponent. Therefore, the integral of \( 3t^2 \) is \( \frac{3t^3}{3} = t^3 \).

The power rule is a quick method to solve polynomial integrals and is widely used in solving calculus problems.

Indefinite Integral

An indefinite integral is a function that represents the collection of all antiderivatives of a given expression. In essence, it is the general solution to a differential equation. Unlike definite integrals, which provide a numerical result, an indefinite integral takes the form of a function.

• It is represented with the integral sign \( \int \) followed by the function you want to integrate, ending with \( + C \).
• The \( + C \) represents an arbitrary constant, as there are infinitely many antiderivatives differing by a constant.

For example, the indefinite integral \( \int (3t^2 + \frac{t}{2}) \, dt \) requires finding antiderivatives for each term:

• The first term, \( 3t^2 \), integrates to \( t^3 \).
• The second term, \( \frac{t}{2} \), integrates to \( \frac{t^2}{4} \).

Combining both gives us the indefinite integral: \( t^3 + \frac{t^2}{4} + C \). This is the most general form of the antiderivative for the given expression.

Constant of Integration

The constant of integration, denoted as \( C \), is an essential part of indefinite integrals. When we integrate a function, there are infinitely many solutions because the derivative of a constant is zero.

• Without \( C \), the solution to the indefinite integral would be incomplete.
• The constant represents all possible vertical shifts of the antiderivative graph.

Therefore, whenever you perform integration, it's crucial to include \( + C \) in your answer.
In the example of \( \int (3t^2 + \frac{t}{2}) \, dt \), the solution is \( t^3 + \frac{t^2}{4} + C \).

• If you differentiate this function, \( \frac{d}{dt} (t^3 + \frac{t^2}{4} + C) \), you return to the original function \( 3t^2 + \frac{t}{2} \) because the derivative of \( C \) is zero.

This demonstrates that the constant \( C \) doesn't affect the validity of the solution but ensures its completeness.