Problem 40 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(x^{3}-4\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{x^4}{4} - 4x + C \).

Step 1: Understand the Integral

The integral \( \int (x^3 - 4) \, dx \) represents finding the antiderivative of the function \( x^3 - 4 \). We will find a function \( F(x) \) whose derivative is \( x^3 - 4 \).

Step 2: Integrate \( x^3 \)

To find the antiderivative of \( x^3 \), apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Thus, the antiderivative of \( x^3 \) is \( \frac{x^{4}}{4} \).

Step 3: Integrate \(-4\)

The integral of a constant \(-4\) is \(-4x\). This is because the derivative of \(-4x\) with respect to \(x\) is \(-4\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to form the antiderivative of the entire expression: \( \int (x^3 - 4) \, dx = \frac{x^4}{4} - 4x + C \), where \( C \) is the constant of integration.

Key Concepts

Antiderivative

An antiderivative is essentially the reverse of taking a derivative. While a derivative tells us the rate at which something changes, an antiderivative gives us the original function before it was differentiated. In the original exercise, we are tasked with finding the antiderivative of the function \( x^3 - 4 \). To do this, we need to find a function whose derivative equals the given function, i.e., \( x^3 - 4 \).

• Antiderivatives are crucial in calculus because they allow us to reverse the process of differentiation.
• Finding an antiderivative involves techniques such as the power rule for integration and dealing with constants, which we'll discuss further.

By solving the problem, we determine the antiderivative as \( \frac{x^4}{4} - 4x + C \), where \( C \) is the constant of integration, ensuring that our solution accounts for all possible functions that differentiate to \( x^3 - 4 \).

Power Rule for Integration

The power rule for integration is a fundamental tool used to find antiderivatives of polynomial terms, like the term \( x^3 \) in our exercise. The rule states that:
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]This means that to integrate a power function, we increase the power by one and divide by the new power.

• For our term \( x^3 \), applying the power rule results in \( \frac{x^4}{4} \).
• The power rule only works when \( n eq -1 \). This is because the function \( x^{-1} \) results in a logarithmic function, which requires a separate integration approach.

Understanding this rule simplifies the process of integration, especially for polynomials, as it directly connects back to taking derivatives.

Constant of Integration

The constant of integration, often denoted as \( C \), plays a vital role in indefinite integrals. When we find the antiderivative of a function, \( C \) represents an unknown constant added to the result. This is because when differentiating, any constant added to a function disappears.

• In the solution \( \int (x^3 - 4) \, dx = \frac{x^4}{4} - 4x + C \), \( C \) accounts for any constant value that could have been part of the original function.
• Including \( C \) makes sure that our antiderivative generalizes to all possible functions that have the same derivative.
• The notion of \( C \) reinforces that indefinite integration provides a family of functions rather than a unique solution.

In conclusion, the constant of integration is about recognizing the infinite possibilities hidden within the differentiated form, and acknowledging them in our integration solutions.