Problem 26 - Thomas Calculus

Question

In Exercises \(17-56,\) 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

\(-4x^{-1/4} + C \)

Step 1: Identify the power rule for integration

To find the antiderivative of a function of the form \( x^n \), we use the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. In this problem, \( n = -\frac{5}{4} \).

Step 2: Apply the power rule to the given function

Apply the power rule to \( x^{-\frac{5}{4}} \): \[ \int x^{-\frac{5}{4}} \, dx = \frac{x^{(-\frac{5}{4})+1}}{-\frac{5}{4}+1} + C = \frac{x^{-\frac{1}{4}}}{-\frac{1}{4}} + C \].

Step 3: Simplify the expression

Simplify the expression by calculating the denominator: Since \( -\frac{1}{4} \) in the denominator, the fraction becomes: \( \frac{-4}{1} \cdot x^{-\frac{1}{4}} = -4x^{-\frac{1}{4}} \). Thus, the antiderivative is \(-4x^{-\frac{1}{4}} + C \).

Step 4: Check by Differentiation

Verify the result by taking the derivative of the antiderivative: \( \frac{d}{dx}(-4x^{-\frac{1}{4}} + C) = -4 \cdot (-\frac{1}{4}) \cdot x^{-\frac{1}{4}-1} = x^{-\frac{5}{4}} \). The derivative matches the original integrand, confirming that the antiderivative is correct.

Key Concepts

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In simpler terms, if you have a function and you find another function such that when you differentiate it, you get your original function, then you have found an antiderivative. For example, since the derivative of \(-4x^{-rac{1}{4}}\) is \(x^{-rac{5}{4}}\), we say that \(-4x^{-rac{1}{4}}+C\) is the antiderivative of \(x^{-rac{5}{4}}\).
Antiderivatives are important for finding indefinite integrals. An indefinite integral of a function is represented by the integral symbol \(\int\), and it produces the general form of antiderivatives, which includes a constant \(C\). This constant is because when we differentiate, the constant disappears, and thus the original function could have been derived from multiple original functions, each differing by a constant.

• Antiderivative involves reversing differentiation.
• Antiderivatives include an arbitrary constant \(C\).
• Finding the antiderivative is the same as finding the indefinite integral.

Power Rule for Integration

The power rule for integration is a method for finding antiderivatives of power functions, which are functions of the form \(x^n\). This rule states that the antiderivative of \(x^n\) is given by \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), provided \(n eq -1\). Using this rule, you simply increase the exponent by 1 and divide by the new exponent, then add the constant \(C\).
In our exercise, the function \(x^{-rac{5}{4}}\) follows the power rule for integration. By setting \(n = -\frac{5}{4}\), we increment the exponent by 1 to get \(-\frac{1}{4}\), divide by this new exponent, and add \(C\). This leads us to the antiderivative \(-4x^{-rac{1}{4}} + C\).

• The power rule is a shortcut for integrating power functions.
• It simplifies the process of finding antiderivatives of polynomial terms.
• Ensure the exponent is not \(-1\) when using this rule.

Differentiation

Differentiation is the process of finding the derivative of a function. In essence, it measures the rate at which the function's value changes. For example, if you have a quantity changing over time, its derivative will tell you how fast it's changing at any particular moment.
The derivative is symbolized by \(\frac{d}{dx}\) or another notation, \(f'(x)\). Differentiation undoes what integration does; hence, checking your antiderivative with differentiation is a great way to verify your work. So, when you find an antiderivative and differentiate it, you should end up back at the original function you started with.
To verify our antiderivative, we differentiate \(-4x^{-\frac{1}{4}} + C\). The derivative \(-4(-\frac{1}{4})x^{-\frac{1}{4}-1}\) simplifies to \(x^{-\frac{5}{4}}\), the original function we started with in the integral. Hence, this confirms our antiderivative is correct.

• Differentiation helps determine the rate of change.
• It is the reverse operation of integration.
• Use differentiation to verify that an antiderivative is correct.