Problem 29 - 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. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

Step-by-Step Solution

Answer

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

Step 1: Identify Components for Integration

The given function inside the integral is \(3t^2 + \frac{t}{2}\). This indicates that we need to find the indefinite integral of each term separately with respect to \(t\).

Step 2: Integrate the First Term

Integrate \(3t^2\) with respect to \(t\). The power rule for integration states that \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\). Applying this, we have: \(\int 3t^2 \, dt = \frac{3t^{3}}{3} = t^3\).

Step 3: Integrate the Second Term

Integrate \(\frac{t}{2}\) with respect to \(t\). This is equivalent to finding \(\int \frac{1}{2}t^1 \, dt\). Using the power rule again, we have: \(\int \frac{1}{2}t \, dt = \frac{1}{2} \cdot \frac{t^2}{2} = \frac{t^2}{4}\).

Step 4: Combine Results

Combine the results of the previous steps to construct the general antiderivative: \(t^3 + \frac{t^2}{4} + C\), where \(C\) is the constant of integration.

Step 5: Verify by Differentiation

Take the derivative of \(t^3 + \frac{t^2}{4} + C\) to confirm correctness. The derivative is \(3t^2 + \frac{2}{4}t = 3t^2 + \frac{t}{2}\), which matches the original integrand, confirming our solution.

Key Concepts

Antiderivative

An antiderivative, sometimes referred to as an indefinite integral, is a function whose derivative returns the original function that you began with. It acts as the reverse process of differentiation. When we're tasked with finding an antiderivative, what we're essentially doing is solving the reverse of the derivative problem.
This is crucial because it helps us understand how functions originally developed from simpler components. In the context of our exercise, the goal is to find the antiderivative of the expression \( 3t^2 + \frac{t}{2} \). By systematically breaking down the given expression, we can find individual antiderivatives for each term and combine them for the general solution.

• Each term in the expression is integrated separately.
• Combining these integrations results in the general antiderivative.

Finding the antiderivative is often aided by rules like the Power Rule, or techniques like Integration by Parts, which we will discuss in further sections.

Integration by Parts

Integration by Parts is a valuable technique used for integrating functions that are products of two more manageable functions. It arises from the product rule for differentiation.
However, in our current exercise, this particular method is not necessary, as the integrals involve simpler terms that can be more efficiently solved by straightforward application of the Power Rule.
Understanding when to apply Integration by Parts is crucial because it often simplifies integrals that seem complex at first glance.Nonetheless, for more complex functions that appear as a product, the formula is given by:\[\int u \, dv = uv - \int v \, du\]- Choose \( u \) and \( dv \) wisely to simplify computation.
- A good rule of thumb is to pick \( u \) from functions that become simpler upon differentiation.For our basic polynomial terms like \( 3t^2 \) and \( \frac{t}{2} \), using the Power Rule suffices, thus avoiding unnecessary complexity.

Power Rule

The Power Rule is a simple yet powerful integration tool. It is specifically used for integrating polynomial expressions, where each term can be expressed in the form \( t^n \). For this rule, if you have a function \( t^n \), its integral is given by:\[ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \]In our exercise, the Power Rule is applied to each term:

• For \( 3t^2 \), integrate to get \( t^3 \).
• For \( \frac{t}{2} \), rewrite as \( \frac{1}{2}t^1 \), then integrate to get \( \frac{t^2}{4} \).

This technique is especially useful for functions that are simple polynomials.An important note: the Power Rule cannot be used for terms where \( n = -1 \); different rules apply in such cases, like logarithmic integration. Nonetheless, for our polynomial terms, it is a straightforward and efficient approach.

Constant of Integration

The Constant of Integration is a key aspect of any indefinite integral. When we find an indefinite integral, it represents a whole family of functions, each differing by a constant value.
The notation \( + C \) is included at the end of our integrated function as a reminder that other similar functions exist which could differ only by this constant. For instance, when we integrated our original expression \( 3t^2 + \frac{t}{2} \), the general antiderivative is \( t^3 + \frac{t^2}{4} + C \).Why is it important?- It reflects the general nature of indefinite integrals, which contain infinitely many solutions.
- Leaving out the constant could imply the incorrect assumption that only one unique function is the antiderivative.Always remember: when integrating to find the antiderivative, the presence of \( + C \) at the end is crucial for representing the complete solution set.