Problem 26 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int x^{-5 / 4} d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \(-4x^{-1/4} + C\).

Step 1: Understanding the Problem

The exercise asks us to find the most general antiderivative, also known as the indefinite integral, of the function \( x^{-5/4} \). This means we need to integrate the function with respect to \( x \).

Step 2: Applying the Power Rule of Integration

The power rule for integration states that for \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). In this case, \( n = -\frac{5}{4} \), thus applying the rule gives: \( \int x^{-5/4} \, dx = \frac{x^{-5/4 + 1}}{-5/4 + 1} + C \).

Step 3: Simplifying the Exponent

Simplify the exponent: \(-\frac{5}{4} + 1 = -\frac{5}{4} + \frac{4}{4} = -\frac{1}{4} \). So the expression becomes \( \frac{x^{-1/4}}{-1/4} + C \).

Step 4: Simplifying the Fraction

The fraction \( \frac{x^{-1/4}}{-1/4} \) can be simplified by multiplying with the reciprocal: \( \frac{x^{-1/4}}{-1/4} = -4x^{-1/4} \). Thus, the most general antiderivative is \( -4x^{-1/4} + C \).

Step 5: Checking the Solution by Differentiation

To verify, differentiate \( -4x^{-1/4} + C \) with respect to \( x \). The derivative is \( -4 \times \left(-\frac{1}{4} x^{-1/4 - 1}\right) = x^{-5/4} \), which matches the original function, confirming the solution.

Key Concepts

Antiderivative

To understand the concept of an *antiderivative*, we can think of it as the reverse process of differentiation. If differentiation is about finding the slope or rate of change of a function, finding the antiderivative means finding a function whose derivative equals the original function. In mathematics, this process is also referred to as finding the indefinite integral. An antiderivative of a function \( f(x) \) is any function \( F(x) \) such that \( F'(x) = f(x) \).

The general form of an antiderivative includes a constant, represented by \( C \). This is because differentiating \( F(x) + C \) would still result in \( f(x) \). This constant is crucial because an infinite number of functions can have the same derivative but differ by a constant. Thus, when we say the "most general antiderivative," it includes this constant, ensuring that no potential solutions are left out.

In our example, the task is to find an antiderivative for the function \( x^{-5/4} \). This process involves integration, which we handle using specific rules, namely integration techniques.

Integration Techniques

Integration techniques are methods used to find the antiderivatives of functions. In our problem, we use a straightforward technique called the power rule of integration. However, other common techniques include substitution, integration by parts, and using trigonometric identities, especially when dealing with more complex functions.

For a function that fits the form of \( x^n \), where \( n eq -1 \), the power rule is typically your go-to technique. However, when functions are more complex, or when they involve trigonometric, exponential, or logarithmic expressions, you may need to use additional methods.

The step-by-step solution showed us how simple algebra allowed us to find the correct antiderivative. But remember, integrating can sometimes require creativity in adjusting forms or using identities to simplify the expression. Additionally, it's important also to think of checking your work by differentiating your final answer to ensure correctness.

Power Rule of Integration

The *power rule of integration* is one of the most fundamental techniques in calculus. It provides an easy way to integrate functions of the form \( x^n \), where \( n eq -1 \). The formula is:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Here's why it's so useful: for any power \( n \) except \(-1\), it gives you a straightforward expression for an antiderivative by increasing the power by 1 and dividing by the new power. The exception \( n eq -1 \) arises because the integral of \( x^{-1} \) is \( \ln|x| + C \), which doesn't follow the same rule.

In this exercise, applying the power rule with \( n = -\frac{5}{4} \), we first adjust the exponent: \(-\frac{5}{4} + 1 = -\frac{1}{4}\). Then divide by the new exponent: \( \frac{x^{-1/4}}{-1/4} \). Simplifying this corresponds to multiplying by \(-4\), leading us to the antiderivative \( -4x^{-1/4} + C \).

Using this rule helps solve a wide variety of problems quickly, making it a crucial component of any calculus toolkit.