Problem 15 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=5 x^{4}+\frac{5}{x^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Antiderivatives

The general antiderivative of a function involves finding a function whose derivative is the original function. This process is also known as integration. We will integrate each term of the given function individually.

Step 2: Integrating the First Term

The first term of the function is \(5x^4\). The antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). Therefore, the antiderivative of \(5x^4\) is \(\frac{5x^{5}}{5} = x^{5}\).

Step 3: Integrating the Second Term

The second term of the function is \(\frac{5}{x^4}\) which can be written as \(5x^{-4}\). The antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), so the antiderivative of \(5x^{-4}\) is \(\frac{5x^{-3}}{-3} = -\frac{5}{3}x^{-3}\).

Step 4: Combine Antiderivatives

Combine the antiderivatives from the previous steps to form the general antiderivative of the given function. The combined form is: \[ x^5 - \frac{5}{3}x^{-3} + C \] where \(C\) is the constant of integration.

Key Concepts

Understanding the Integration Process

To find the antiderivative of a function, we essentially reverse the process of differentiation. This is called integration. While differentiation gives us the rate at which something changes, integration helps us find the overall amount or accumulation. Imagine it like putting back together a broken puzzle where differentiation was how it broke. The antiderivative of a function is a new function, whose derivative will give you the original function again.

• The process involves working term by term.
• Each term of the original function is tackled individually.
• We apply the same formula thanks to the linearity of integration.

These steps make the integration process a systematic one, ensuring each piece correctly forms the whole function with all necessary constants included.

Handling Powers of x

One of the fundamental aspects of integration you must grasp is how we deal with powers of x. Each term in our original function is a polynomial term, meaning it has x raised to some power. The antiderivative process involves increasing the power by one and dividing by this new power.
Here's how we handle each power of x:

• When you have a term like \(x^n\), the antiderivative is found by raising the power to \(n+1\) and dividing by \(n+1\).
• For example, integrating \(5x^4\) results in \(x^5\), as you get \(\frac{5x^5}{5}\).
• For negative exponents such as \(x^{-4}\), the same principle applies. You end up with \(-\frac{5}{3}x^{-3}\) for the term \(5x^{-4}\).

Thus, thoroughly understanding this power of x method is crucial, making it easier to find antiderivatives efficiently.

Significance of the Constant of Integration

When integrating, the constant of integration, represented by \(C\), plays a crucial role. This constant accounts for all vertical shifts that the original function might have experienced. Since differentiation eliminates constants, we can’t pinpoint their original value without additional conditions or initial values.
The constant of integration ensures:

• The general solution reflects all possible particular solutions.
• That the family of antiderivatives shares the same derivative.
• Every potential function that was differentiated into the original function is covered.

Thus, the constant of integration is a vital part of integration, providing the completeness needed for general antiderivatives, ensuring nothing is missed.