Problem 45 - Single Variable Calculus

Question

In Exercises \(45-50\), find a power series representation for the indefinite integral. \(\int \frac{1}{1+x^{3}} d x\)

Step-by-Step Solution

Answer

The power series representation for the indefinite integral \(\int \frac{1}{1+x^3} dx\) is: \[C + x - \frac{1}{4}x^4 + \frac{1}{7}x^7 - \frac{1}{10}x^{10} + \frac{1}{13}x^{13} - \cdots\]

Step 1: Apply Geometric Series Formula to the Integrand Function

We want to find a power series representation for the function \(\frac{1}{1+x^3}\). This looks similar to the geometric series formula \(1 + r + r^2 + ... = \frac{1}{1-r}\) for \(|r| < 1\). Let's make a substitution to rewrite the integrand function in the form of a geometric series. We can rewrite the integrand function by letting \(r = -x^3\): \[\frac{1}{1+x^3} = \frac{1}{1-(-x^3)}\] Now we can apply the geometric series formula for \(|r| < 1\) to find the power series representation of the integrand function: \[\frac{1}{1-(-x^3)} = 1 - x^3 + x^6 - x^9 + x^{12} - \cdots\]

Step 2: Integrate the Power Series Term by Term

We can now integrate the power series representation of the integrand function term by term: \[\int \frac{1}{1+x^3} dx = \int (1 - x^3 + x^6 - x^9 + x^{12} - \cdots) dx\] Integrating term by term, we get: \[C + \int 1 dx - \int x^3 dx + \int x^6 dx - \int x^9 dx + \int x^{12} dx - \cdots\] \[C + x - \frac{1}{4}x^4 + \frac{1}{7}x^7 - \frac{1}{10}x^{10} + \frac{1}{13}x^{13} - \cdots\] Here, C is the constant of integration. Thus, the power series representation for the indefinite integral \(\int \frac{1}{1+x^3} dx\) is: \[C + x - \frac{1}{4}x^4 + \frac{1}{7}x^7 - \frac{1}{10}x^{10} + \frac{1}{13}x^{13} - \cdots\]

Key Concepts

Indefinite Integral

An indefinite integral is the opposite of differentiation. It allows us to find the original function from its derivative. When you integrate a function, you obtain a family of functions plus a constant, because the derivative of a constant is zero. In mathematical notation, if you have a function \(f(x)\), the indefinite integral is represented as \(\int f(x) \, dx = F(x) + C\), where \(F(x)\) is the antiderivative, and \(C\) is the constant of integration.
In the exercise, the goal was to express the indefinite integral \(\int \frac{1}{1+x^3} \, dx\) in terms of a power series. This power series representation will make it easier to work with the function, especially if you need to approximate values within a specific interval. Remember that when you're dealing with an indefinite integral, adding the constant of integration \(C\) is crucial.
Understanding indefinite integrals is key when you want to reverse the process of differentiation and reconstruct a function from its derivative. This principle is at the foundation of integral calculus.

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed ratio. Its general form is \(1 + r + r^2 + r^3 + \cdots = \frac{1}{1-r}, \quad |r| < 1\). This formula applies when the terms of the series are decreasing in magnitude due to the constraint \( |r| < 1 \), ensuring convergence.
In the context of this problem, we utilized the geometric series to rewrite \(\frac{1}{1+x^3}\) as a power series. By choosing \( r = -x^3 \), this transforms our function into the series \(1 - x^3 + x^6 - x^9 + \cdots\). This step was crucial in translating the original function into a form that could be integrated term by term.
Understanding and applying the geometric series is essential for breaking down complex functions into simpler, more manageable parts. This technique is especially valuable in calculus when approximating functions over intervals.

Term by Term Integration

Term by term integration is a method used to integrate series, where you integrate each term of the series separately. This approach is particularly useful when the series representation of a function, like a power series, can be easily integrated term by term.
In the exercise, after transforming the integrand \(\frac{1}{1+x^3}\) into its power series form, each term was integrated individually. For example, \(\int 1 \, dx = x\), \(\int x^3 \, dx = \frac{1}{4} x^4\), and so on. The integration results in the power series \(C + x - \frac{1}{4}x^4 + \frac{1}{7}x^7 - \cdots\), showing how each term contributes to the final integral.
This method simplifies the integration process and helps when dealing with functions that can be challenging to integrate directly. It's important to note that term by term integration works well for convergent power series. This approach highlights the elegance of calculus in decomposing complex problems into simpler steps.