Problem 20 - 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. $$\int\left(\frac{t^{2}}{2}+4 t^{3}\right) d t$$

Step-by-Step Solution

Answer

\( \frac{t^3}{6} + t^4 + C \) is the most general antiderivative.

Step 1: Break Down the Integral

The integral given is: \[ \int \left(\frac{t^{2}}{2} + 4t^{3} \right) dt \]We need to find the antiderivative for each term separately. Split the integral:\[ \int \frac{t^{2}}{2} \, dt + \int 4t^{3} \, dt \]

Step 2: Integrate the First Term

The first term is \( \int \frac{t^{2}}{2} \, dt \). Recall the power rule for integration, which states that \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \).Use this to integrate \( \frac{t^2}{2} \):\[ \int \frac{t^2}{2} \, dt = \frac{1}{2} \cdot \frac{t^{3}}{3} = \frac{t^3}{6} \]

Step 3: Integrate the Second Term

The second term is \( \int 4t^{3} \, dt \).Again using the power rule:\[ \int 4t^{3} \, dt = 4 \cdot \frac{t^{4}}{4} = t^4 \]

Step 4: Combine the Results and Include Constant of Integration

Combine the results from the two integrals found in the previous steps:\[ \frac{t^3}{6} + t^4 + C \]where \( C \) is the constant of integration.

Step 5: Verify by Differentiation

Differentiate \( \frac{t^3}{6} + t^4 + C \) to confirm it is the correct antiderivative.\[ \frac{d}{dt} \left( \frac{t^3}{6} + t^4 + C \right) = \frac{3t^2}{6} + 4t^3 = \frac{t^2}{2} + 4t^3 \]This matches the original function, confirming that our solution is correct.

Key Concepts

Indefinite Integral

The indefinite integral is a fundamental concept in calculus. Unlike definite integrals, which calculate the area under a curve within specific bounds, indefinite integrals find a family of functions. These functions are antiderivatives of the original function, meaning their derivative returns the integrand. For example, when we are given an expression like \( \int \left( \frac{t^2}{2} + 4t^3 \right) dt \), we are essentially trying to find a function whose derivative will bring us back to \( \frac{t^2}{2} + 4t^3 \).

• Represents a family of functions, all differing by a constant (the constant of integration).
• The process involves finding the antiderivative of a given

Power Rule

The power rule is a straightforward yet highly essential mechanism in calculus for finding antiderivatives or indefinite integrals. The rule is summarized by the formula \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), where \( n \) is any real number other than -1 and \( C \) represents the constant of integration.The power rule greatly simplifies the integration process:

• Increase the exponent by 1.
• Divide by the new exponent.
• Add the constant of integration \( C \) to complete the process.

In the provided exercise, for the term \( \int \frac{t^2}{2} \, dt \), we use the power rule by first taking care of the constant \( \frac{1}{2} \) and then applying the rule to \( t^2 \), which leads us to \( \frac{t^3}{6} \). Again, for the term \( \int 4t^3 \, dt \), we apply the power rule after factoring out the constant 4, arriving at \( t^4 \). The power rule provides a systematic way to solve integrals that adhere to polynomial expressions.

Constant of Integration

The constant of integration, denoted as \( C \), is perhaps one of the most significant aspects of indefinite integrals. It's crucial because it embodies the idea that antiderivatives are not unique.

• Every function has an infinite number of antiderivatives, as they can differ by any constant.
• When calculating indefinite integrals, always include this constant to indicate the family of functions.

In our solution, while combining individual antiderivatives, we include the constant \( C \). This step ensures the solution is comprehensive and accurate. The inclusion of \( C \) is vital for applications that involve solving differential equations, where initial or boundary conditions help determine the specific value of \( C \). By acknowledging \( C \), we affirm the complete solution, acknowledging both known and unknown possibilities for the original function.