Problem 6

Question

Find all antiderivatives of the given function. $$ 4 x^{2}+6 x-1 $$

Step-by-Step Solution

Verified

Answer

The antiderivatives are $F(x) = \frac{4}{3}x^3 + 3x^2 - x + C$.

1Step 1: Understand the Problem

We are asked to find all the antiderivatives of the function $f(x) = 4x^2 + 6x - 1$. The antiderivative is also known as the indefinite integral, which means we need to find a function $F(x)$ such that $F'(x) = f(x)$.

2Step 2: Apply the Power Rule for Integration

Recall that the power rule for integration states that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$, provided $n eq -1$. We will apply this rule to each term of the polynomial separately.

3Step 3: Integrate Each Term Separately

Start integrating each term of the polynomial: - The antiderivative of $4x^2$ is $\frac{4}{3}x^3$. - The antiderivative of $6x$ is $3x^2$. - The antiderivative of $-1$ is $-x$.

4Step 4: Combine the Antiderivatives

Combine the antiderivatives of each term to form the complete antiderivative:\[ F(x) = \frac{4}{3}x^3 + 3x^2 - x + C \]where $C$ is the constant of integration.

5Step 5: Final Verification

Differentiate the function $F(x) = \frac{4}{3}x^3 + 3x^2 - x + C$ to ensure it equals the original function $f(x) = 4x^2 + 6x - 1$. Differentiating term by term gives $4x^2 + 6x - 1$, which matches $f(x)$, confirming our solution is correct.

Key Concepts

Indefinite IntegralPower Rule for IntegrationPolynomial Integration

Indefinite Integral

In calculus, the indefinite integral is a fundamental concept you will encounter often. It involves finding a function whose derivative matches a given function. Basically, if you have a function $ f(x) $, you want to find another function $ F(x) $ such that when you take its derivative, you get back to $ f(x) $. This $ F(x) $ is called the antiderivative. An indefinite integral is expressed using the integral symbol $ \int $ and signifies that you are looking for a family of functions rather than a single one. The general form is:

$ \int f(x) \, dx = F(x) + C $

Here, $ C $ is the constant of integration. This constant reflects the fact that there are infinitely many antiderivatives for each function, differing only by this constant. When you see an indefinite integral, remember it's all about reversing differentiation and finding that mysterious function that it derived from.

Power Rule for Integration

The power rule for integration comes to the rescue when dealing with polynomials, making the process of finding antiderivatives a lot easier. It's one of the first rules you learn when you set off on your integration journey.When applying the power rule, for any term $ x^n $, the antiderivative is given by:

$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $

This rule applies as long as $ n eq -1 $. The idea is to increase the power by one and then divide by that new power. It's straightforward and quite handy for polynomials where each term can be directly integrated using this rule.Let's revisit the function $ f(x) = 4x^2 + 6x - 1 $. By applying the power rule:

The term $ 4x^2 $ becomes $ \frac{4}{3}x^3 $
The term $ 6x $ becomes $ 3x^2 $
And a constant like $ -1 $ becomes $ -x $

Thus, each term is integrated individually using this simple yet powerful rule.

Polynomial Integration

Polynomial integration involves finding the antiderivative of a polynomial function, which is essentially a sum of powers of $ x $. Luckily, with tools like the power rule for integration, tackling polynomial integration becomes manageable. For the polynomial $ f(x) = 4x^2 + 6x - 1 $, apply the power rule to each term independently:First, deal with each term one at a time:

$ 4x^2 \rightarow \frac{4}{3}x^3 $
$ 6x \rightarrow 3x^2 $
$-1 \rightarrow -x $

Next, combine these individual antiderivatives to form the complete solution:

$ F(x) = \frac{4}{3}x^3 + 3x^2 - x + C $

The constant $ C $ represents any constant value and ensures we account for the whole family of antiderivatives. Polynomial integration requires taking each term, applying the power rule, and recompiling the separate results. It is an invaluable skill used in calculus for solving complex problems involving polynomials.

Problem 6

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