Problem 2 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=5\left(1-x^{2}\right) $$

Step-by-Step Solution

Answer

The general antiderivative is \( \int f(x) \, dx = 5x - \frac{5}{3}x^3 + C, \) where \( C \) is a constant.

Step 1: Understand the Function

We need to find the general antiderivative, or indefinite integral, of the function given by \[ f(x) = 5(1-x^2). \]This requires us to apply the rules of integration to each part of the expression inside the function.

Step 2: Separate the Function

Identify the separate terms in the expression. Distribute the factor of 5 across the parentheses:\[ f(x) = 5 \times 1 - 5 \times x^2 = 5 - 5x^2. \]Now we have two terms that we will integrate separately.

Step 3: Integrate Each Term

Now, integrate each term separately using basic integration rules. The integral of a constant \(5\) with respect to \(x\) is \(5x\). The integral of \(-5x^2\) with respect to \(x\) is \[-\frac{5}{3}x^{3},\] since the power rule for integration states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).

Step 4: Combine the Integrals

Add the results from integrating each term:\[ \int f(x) \, dx = 5x - \frac{5}{3}x^3 + C, \]where \(C\) is the constant of integration that comes from the indefinite integral. This is the general form of the antiderivative.

Step 5: Verify the Antiderivative

Differentiate the antiderivative to verify that it produces the original function:\[ \frac{d}{dx}(5x - \frac{5}{3}x^3 + C) = 5 - 5x^2. \]Since the result matches the original function \(f(x)\), our antiderivative is correct.

Key Concepts

Integration

Integration is a core aspect of calculus, often described as the reverse process of differentiation. While differentiation involves finding the rate of change, integration involves finding the original function given its derivative. This means it helps us find functions when we know their rate of change, like measuring the area under curves on a graph.
The process of integration allows us to add up small quantities, which can be thought of as a summation of tiny parts to find a whole. This technique is applied in physics to calculate quantities like mass, area, and volume. In mathematics, it can be used to solve various functional equations. One of the most fundamental methods used in integration is applying specific rules which help simplify and solve integrals for different forms of functions.
For any given function, integration helps us determine an antiderivative, providing a way to recover the function that was initially differentiated.

Indefinite Integral

The term indefinite integral refers to finding an entire family of antiderivatives for a function. Instead of producing a numeric answer like in definite integrals, indefinite integrals yield a function, expressed in terms of a variable plus a constant. This constant, often denoted as "C," is crucial because it represents any constant that, when differentiated, would disappear (since the derivative of a constant is zero).
In practice, computing the indefinite integral involves applying integration techniques to each part of a function separately and then summing the results. The result is expressed as an antiderivative with the constant of integration. To verify the correctness, we can differentiate this antiderivative. If our function reappears, our indefinite integral computation is correct.
Indefinite integrals are fundamental in real-world applications like computing displacement from velocity or reconstructing original price functions from cost functions.

Power Rule for Integration

The power rule for integration is a corner-stone technique used to integrate terms of the form \(x^n\). This rule simplifies integrating polynomial functions.
The power rule states that the integral of \(x^n\) with respect to \(x\) is \( \frac{x^{n+1}}{n+1} + C \), provided that \(neq -1\). The addition of \(C\), the constant of integration, acknowledges the family of functions that differ only by a constant.
Applying the power rule requires careful attention to the exponent \(n\):

• Increase the power of \(x\) by one, yielding \(x^{n+1}\).
• Divide by the new power \(n+1\).
• Add the constant of integration \(C\).

In practical scenarios, this rule transforms complex polynomial expressions into integrable forms, making it easier to derive general antiderivatives. This essential tool is employed whenever determining the antiderivatives of polynomial functions to reconstruct original functions from their derivatives.