Problem 6 - Calculus for Biology and Medicine

Question

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2 x^{3}+x^{2}-5 x $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{1}{2}x^4 + \frac{1}{3}x^3 - \frac{5}{2}x^2 + C \).

Step 1: Identify the Function Components

The given function is \( f(x) = 2x^3 + x^2 - 5x \). We will find the antiderivative by integrating each term separately.

Step 2: Antiderivative of each Term

Use the power rule for integration, which states that the antiderivative of \( ax^n \) is \( \frac{a}{n+1}x^{n+1} \). Applying this to each term:1. The antiderivative of \( 2x^3 \) is \( \frac{2}{3+1}x^{3+1} = \frac{1}{2}x^4 \).2. The antiderivative of \( x^2 \) is \( \frac{1}{2+1}x^{2+1} = \frac{1}{3}x^3 \).3. The antiderivative of \( -5x \) is \( \frac{-5}{1+1}x^{1+1} = -\frac{5}{2}x^2 \).

Step 3: Combine the Antiderivatives

Add the antiderivatives of each term to form the general antiderivative of the function:\[ F(x) = \frac{1}{2}x^4 + \frac{1}{3}x^3 - \frac{5}{2}x^2 + C \] where \( C \) is the constant of integration.

Step 4: Write the Final Answer

The general antiderivative of the function \( f(x) = 2x^3 + x^2 - 5x \) is:\[ F(x) = \frac{1}{2}x^4 + \frac{1}{3}x^3 - \frac{5}{2}x^2 + C \]

Key Concepts

Integration

Integration is a fundamental concept in calculus, serving as the reverse process of differentiation. It's the powerful tool we use to find antiderivatives. In simple terms, integration helps us determine the original function given its derivative. Here’s why it’s crucial:

• Integration can compute areas under curves, which is invaluable in physics and engineering for solving real-world problems.
• Besides area, it can also find accumulated quantities like distance from velocity.
• The integral's outcome forms the general antiderivative of a function.

In the given exercise, integration is applied to each term of the polynomial function individually. By integrating, we effectively "add up" infinitely small pieces to uncover the original function's form. This leads directly to finding the general antiderivative, a key feature of the exercise task.

Power Rule

The power rule for integration is a straightforward technique crucial for finding antiderivatives of polynomial functions. It provides a formula to transform each term of the function individually. Here's how the power rule works:

• The rule states that to integrate a term of the form \( ax^n \), the antiderivative is \( \frac{a}{n+1}x^{n+1} \).
• It requires the exponent \( n \) to be different from \(-1\), as this case relates to logarithmic functions.
• This rule simplifies integration by making it a predictable and simple operation.

In our problem, each term of the function \( f(x) = 2x^3 + x^2 - 5x \) was resolved separately utilizing this rule:

• For \( 2x^3 \), applying the power rule yields \( \frac{1}{2}x^4 \).
• For \( x^2 \), it becomes \( \frac{1}{3}x^3 \).
• And \( -5x \) results in \( -\frac{5}{2}x^2 \).

Each antiderivative term fits neatly into the final expression, demonstrating the elegance and efficiency of the power rule.

Constant of Integration

When we integrate a function, there is an important element referred to as the constant of integration, denoted as \( C \). This constant acknowledges that there are infinitely many antiderivatives for a given function. Here’s why it's essential:

• The constant accounts for any vertical shifts from the original function, as different functions can have the same derivative.
• In practical applications, initial conditions or additional information are used to solve for \( C \).
• Without \( C \), a solution to the integral is considered incomplete.

In the solution to the given problem, after applying the power rule and integrating each term, the constant of integration \( C \) was added to the result, \( F(x) = \frac{1}{2}x^4 + \frac{1}{3}x^3 - \frac{5}{2}x^2 + C \). This ensures the general antiderivative is comprehensive and flexible for any specific solution when required conditions are provided.