Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int \frac{x^{4}}{1-x^{5}} d x$$

Step-by-Step Solution

Answer

The indefinite integral of the function \(\frac{x^4}{1-x^5}\) with respect to x is \(-\frac{1}{5}\ln{|1-x^5|} + C\), where C is the constant of integration.

Step 1: Identify the substitution

We notice that the denominator is almost the derivative of the numerator, with only a sign difference. Therefore, let's make a substitution: Set \(u = 1 - x^5\) Then, take the derivative with respect to x, which gives us: \(d u = -5x^4 d x\)

Step 2: Rewrite the integral in terms of u

Using the substitution from step 1 (with rearranged variables), we can rewrite the integral in terms of u: \(-\frac{1}{5}\int \frac{d u}{u}\)

Step 3: Integrate with respect to u

The integral we now have is a simple one. It's the integral of a function of the form \(\frac{1}{u}\), which is just the natural logarithm. Thus, we get: \(-\frac{1}{5}\ln{|u|} + C\) where C is the constant of integration.

Step 4: Substitute back x

Since we are interested in the integral with respect to x, we need to substitute back x into the expression we got in step 3. From our original substitution, we have \(u = 1 - x^5\), so the final indefinite integral is: \(-\frac{1}{5}\ln{|1-x^5|} + C\)

Key Concepts

U-Substitution

The technique of u-substitution is a method for finding indefinite integrals and is analogous to the chain rule for differentiation. In the context of integration, it involves choosing a new variable, typically denoted as 'u', which is a function of the original variable, and then rewriting the integral in terms of 'u'. This simplifies the problem into a form that can be more easily integrated.

When employing u-substitution, it is crucial to correctly identify the part of the integrand that will simplify the integral. A good target for substitution is a function whose derivative appears elsewhere in the integrand. In our example, we set \( u = 1 - x^5 \), because the derivative of \( -x^5 \) is \( -5x^4 \), which corresponds to the numerator, save for the constant factor. Correctly executing u-substitution often requires adjusting for constants, as we did by factoring out the \( -1/5 \) in step 2.

Integration Techniques

Various strategies exist for integrating functions, and each technique is useful for different kinds of functions. Besides u-substitution, other techniques include integration by parts, partial fractions decomposition, trigonometric integrals, and integration by trigonometric substitution. Knowing which method to apply depends on recognizing patterns in the integrand.

In our example, u-substitution is the appropriate technique, but for more complex functions, it might necessitate combining several methods. Understanding how to manipulate expressions to fit the pattern of a known integral is key. In some cases, multiple approaches could work, and it's often a matter of personal preference or instructional guidance as to which method is used.

Natural Logarithm

The natural logarithm, denoted as \( \text{ln}(x) \), is the inverse operation to raising the number 'e' (an irrational and transcendental number approximately equal to 2.71828) to a power. The derivative of \( \text{ln}(x) \) with respect to 'x' is \( 1/x \), and consequently, the indefinite integral of \( 1/u \) with respect to 'u' is \( \text{ln}|u| + C \), where 'C' represents the constant of integration.

In our problem, after substituting and rewriting the integral, we quickly identify that the integrand is in the form that leads to the natural logarithm, hence we end with \( -\frac{1}{5}\text{ln}|u| + C \). It's important to remember that the argument of the natural logarithm must always be positive, which is why we use absolute value signs around 'u'.