Problem 28 - University Calculus: Early Transcendentals

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

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

Step 1: Distribute the Integral

Start by distributing the integral over each term separately. We have \( \int \frac{t^2}{2} \, dt + \int 4t^3 \, dt \). This allows us to find the antiderivative of each term individually.

Step 2: Find Antiderivative of First Term

The antiderivative of \( \frac{t^2}{2} \) with respect to \( t \) is found by increasing the exponent by 1 and dividing by the new exponent: \( \frac{t^3}{3 \cdot 2} = \frac{t^3}{6} \).

Step 3: Find Antiderivative of Second Term

The antiderivative of \( 4t^3 \) is similarly found by increasing the exponent by 1 and dividing by the new exponent: \( 4 \cdot \frac{t^{4}}{4} = t^{4} \).

Step 4: Combine the Antiderivatives and Add Constant of Integration

Combine the antiderivatives of each term to get the general antiderivative: \( \frac{t^3}{6} + t^4 + C \), where \( C \) represents the constant of integration.

Step 5: Verify by Differentiation

Differentiate the result \( \frac{t^3}{6} + t^4 + C \). Applying differentiation term by term, we get: \( \frac{d}{dt} \left( \frac{t^3}{6} \right) = \frac{t^2}{2} \), \( \frac{d}{dt} \left( t^4 \right) = 4t^3 \), and \( \frac{d}{dt} (C) = 0 \). This confirms the solution as it matches the original integrand \( \frac{t^2}{2} + 4t^3 \).

Key Concepts

Indefinite Integral

An indefinite integral is a fundamental concept in calculus that revolves around finding the most general form of an antiderivative for a given function. When you see an expression like \( \int f(t) \, dt \), you're being asked to determine a function whose derivative gives you back the original function \( f(t) \). This is what we call the process of integration.

Unlike a definite integral, which calculates the area under a curve between two fixed points, an indefinite integral provides a family of functions, each differing by a constant term. This lack of boundaries is why it’s called "indefinite." The result is a function plus a constant of integration, denoted as \( C \).

For example, finding the indefinite integral of \( \int \left( \frac{t^{2}}{2} + 4t^{3} \right) \, dt \) involves distributing the integral over each term, then calculating the antiderivative of each term separately. In our solution, these are \( \frac{t^3}{6} \) and \( t^4 \). Once found, these antiderivatives are combined to form a single expression, which leads us to the most general function that can be derived from the original expression.

Constant of Integration

When dealing with indefinite integrals, one of the key elements we must consider is the 'constant of integration,' often represented by the letter \( C \). This constant represents the idea that there are many possible antiderivatives for a given function, differing only by a constant value.

Since the derivative of any constant is zero, any constant added to an antiderivative will vanish when differentiated, making it impossible to detect its original presence solely through differentiation. Therefore, whenever you find an indefinite integral, you always add \( C \) to your result. This ensures completeness, acknowledging that there are infinitely many such functions that satisfy the criteria.

In our exercise, when combining the antiderivatives \( \frac{t^3}{6} + t^4 \), we add \( C \) to express the most general solution possible, which accounts for all vertical shifts of the integral function without altering its derivative.

Differentiation

Differentiation is the mathematical process of finding the rate at which a function is changing at any given point. It's essentially the reverse operation of integration, helping us verify the results of our integration work.

To ensure that the antiderivative you've found is correct, you differentiate it to check if it recovers the original function you started with. This method is essential in confirming the accuracy of the integration.

For the exercise \( \int \left( \frac{t^{2}}{2} + 4t^{3} \right) \, dt \), after determining the antiderivative \( \frac{t^3}{6} + t^4 + C \), differentiating each term should lead you back to \( \frac{t^{2}}{2} + 4t^{3} \). In this sense, differentiation acts as a crucial checkpoint, ensuring that the methods used in integration provide a valid solution. By checking each term, from \( \frac{d}{dt} \left( \frac{t^3}{6} \right) = \frac{t^2}{2} \) and \( \frac{d}{dt} \left( t^4 \right) = 4t^3 \), to \( \frac{d}{dt} (C) = 0 \), we verify our solution accurately matches the original integrand.