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

\( \frac{x^4}{4} - 4x + C \)

Step 1: Identify the Components of the Integral

The integral to be solved is \( \int\left(x^{3}-4\right) d x \). This is a polynomial function with two terms: \( x^3 \) and \(-4\). We need to integrate each term separately.

Step 2: Integrate Each Term Separately

For a function \( x^n \), the integral is \( \frac{x^{n+1}}{n+1} \). Here, we integrate \( x^3 \) to get \( \frac{x^{4}}{4} \). The integral of a constant \( -4 \) is \( -4x \). So, we have \( \int x^3 \,dx = \frac{x^4}{4} \) and \( \int -4 \, dx = -4x \).

Step 3: Combine the Results and Add Constant of Integration

Combine the results from the previous step. The integral is \( \frac{x^4}{4} - 4x + C \), where \( C \) is the constant of integration.

Key Concepts

Polynomial Integration

Polynomial integration involves finding the antiderivatives of polynomial functions. Polynomials are expressions that consist of variables raised to powers and multiplied by coefficients. When integrating polynomials:

• Integrate each term separately.
• Use the rule: add one to the exponent and divide by the new exponent for terms like \( x^n \).

For example, in the given problem \( \int (x^3 - 4) \, dx \), we treat \( x^3 \) and \(-4\) as separate. The integral of \( x^3 \) is \( \frac{x^4}{4} \) and of \(-4\) is \( -4x \). The constant rule states that the integral of a constant \( k \) is \( kx \). Combining these results gives the integral's antiderivative, \( \frac{x^4}{4} - 4x + C \), where \( C \) stands for the constant of integration. This technique simplifies the process of integrating polynomials.

Integration by Parts

Integration by parts is a technique used for integrating products of functions. While this method is not directly applied in the provided exercise, it is an essential tool in calculus, allowing integration of more complex functions.The formula for integration by parts is:\[ \int u \, dv = uv - \int v \, du \]Where:

• \( u \) is a function chosen to differentiate.
• \( dv \) is the remaining part of the integral, to integrate.

Choose \( u \) and \( dv \) based on what simplifies the result. In polynomial terms, you would adjust the choice of \( u \) to reduce the polynomial degree. Though not applied here, this method is critical for higher complexity integrals involving products of polynomial and non-polynomial functions.

Calculus for Biology

Calculus is often applied in biology to model various phenomena. Integration, part of calculus, helps in understanding growth patterns, rates of change, and accumulation processes. In the context of biology:

• Polynomials like growth curves model population changes or biochemical reactions.
• Integration helps compute total populations or concentrations over time.

Considering our example, integrating polynomials is the first step. For instance, if \( x^3 - 4 \) represented a biological process, the integral \( \frac{x^4}{4} - 4x + C \) might indicate the total change in population or chemical concentration over a specific interval. Understanding this integration technique is useful for biologists aiming to transfer calculus skills to biological contexts.