Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{5 / 4} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{4}{9}x^{9/4} + C \).

Step 1: Identify the Function to Integrate

The function given is \( f(x) = x^{5/4} \). We want to find its antiderivative.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \). Here, \( n = \frac{5}{4} \).

Step 3: Calculate the Antiderivative

Using the power rule, increase the exponent by one: \( \frac{5}{4} + 1 = \frac{9}{4} \). Now apply the rule: the antiderivative of \( f(x) = x^{5/4} \) is \( \frac{x^{9/4}}{9/4} + C \).

Step 4: Simplify the Expression

Simplify the expression obtained: \( \frac{x^{9/4}}{9/4} \) can be rewritten as \( \frac{4}{9}x^{9/4} + C \) by multiplying the numerator by the reciprocal of the fraction in the denominator.

Key Concepts

Antiderivatives

In calculus, finding an antiderivative means determining a function whose derivative gives the original function. This process is also known as integration, and it's the reverse operation of differentiation. When we talk about finding the general antiderivative of a function, we refer to a family of functions, all differing by a constant, often represented as \( C \).

• For an antiderivative of a function \( f(x) \), you may encounter it expressed as \( F(x) + C \).
• The constant \( C \) represents any fixed number since when you take a derivative, constants vanish.

Understanding antiderivatives is crucial because it forms the basis of solving differential equations, calculating definite integrals, and understanding the accumulation of quantities represented by integrals.

Power Rule for Integration

The Power Rule for Integration is a straightforward guideline used to find antiderivatives of power functions, which are of the form \( x^n \). This rule allows you to integrate without needing more complex techniques.

• According to this rule, if you have a function \( x^n \), its antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
• This rule simply involves increasing the exponent by one, then dividing by the new exponent.

This simplicity is what makes the power rule essential for introductory calculus problems, especially when direct application is possible, just like in our given problem with the function \( f(x) = x^{5/4} \). The antiderivative is found by raising the exponent by one from 5/4 to 9/4 and then dividing by this new exponent.

Exponents in Integration

When dealing with exponents in integration, it's essential to carefully manage the arithmetic involving fractional (or even negative) exponents. This is key when applying the power rule and simplifying the results.

• Fractional exponents arise frequently, especially in roots and polynomial expressions. Here, we deal with \( x^{5/4} \), a common kind of fractional exponent.
• We must handle these exponents carefully: when incrementing the exponent by one, ensure precision in adding fractions (e.g., \( \frac{5}{4} + \frac{4}{4} = \frac{9}{4} \)).

Once the power is increased, simplifying the coefficients involved in division, such as checking that \( \frac{x^{9/4}}{9/4} \) becomes \( \frac{4}{9} x^{9/4} \), is crucial for clarity and accuracy. As such, correctly handling exponents and their arithmetic paves the way for successful integration.