Problem 5 - Calculus

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: Understand the Problem

We need to find the indefinite integral, or antiderivative, of the given function \(f(x) = x^{5/4}\). This involves finding a function \(F(x)\) such that its derivative \(F'(x)\) equals \(f(x)\).

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n eq -1\). Here, \(n = \frac{5}{4}\).

Step 3: Calculate the Indefinite Integral

Substitute \(n = \frac{5}{4}\) into the power rule formula: \[\int x^{5/4} \, dx = \frac{x^{5/4 + 1}}{5/4 + 1} + C\] Simplify the exponent: \(5/4 + 1 = 9/4\). Therefore, \[\int x^{5/4} \, dx = \frac{x^{9/4}}{9/4} + C\]

Step 4: Simplify the Result

To simplify \(\frac{x^{9/4}}{9/4}\), multiply by the reciprocal of \(\frac{9}{4}\): \[\frac{4}{9}x^{9/4} + C\] Thus, the general antiderivative is \(F(x) = \frac{4}{9}x^{9/4} + C\).

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental concept in calculus that enables us to find antiderivatives of power functions. It's one of the simplest integration rules, making it an essential tool in your problem-solving toolkit. The rule can be stated as follows:

• For any real number \(n\) where \(n eq -1\), the integral of \(x^n\) is given by: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
• \(C\) represents the constant of integration, which is crucial because antiderivatives are not unique; they differ by a constant.

To apply this rule, simply add 1 to the exponent of \(x\) and then divide by the new exponent, remembering to include the constant \(C\). It’s important to note that this rule cannot be used if \(n = -1\). This is because the integral of \(x^{-1}\) (or \(1/x\)) is a special case, leading to the natural logarithm function \(\ln|x|\). Understanding and utilizing the Power Rule properly can simplify many integration problems, as demonstrated in the original exercise.

Indefinite Integral

The indefinite integral, also known as the antiderivative, of a function represents a family of functions whose derivatives are equal to the original function. Unlike a definite integral, which yields a numeric value representing area under a curve, an indefinite integral results in a general expression plus a constant \(C\).
This constant arises because integration is essentially the reverse process of differentiation.

• If you were to differentiate an indefinite integral, it should recover the original function.
• The constant \(C\) represents all potential shifts vertically along the y-axis that don't affect the slope (derivative) of the function.
• Essentially, it captures the notion that there can be infinitely many antiderivatives, each varying by a constant.

In the exercise, we were tasked with finding the indefinite integral of \(f(x) = x^{5/4}\). By following integration rules, particularly the Power Rule, we determined:\[ F(x) = \frac{4}{9}x^{9/4} + C \]This solution effectively covers the entire family of functions that differentiate back to \(f(x) = x^{5/4}\).

Integration

Integration is a core component of calculus that is the reverse operation of differentiation. While differentiation involves finding the rate of change, integration focuses on finding the total accumulation or area under a curve.
In practical terms, integration can be used to calculate:

• Areas under curves or between lines and curves
• Volumes under surfaces
• Central values such as centers of mass or centroids
• Total accumulated quantities in physics, such as distance, mass, or charge

There are two primary types of integration:

• Indefinite Integration, which leads to finding antiderivatives along with a constant of integration
• Definite Integration, where you calculate the exact area under a curve within specific bounds, providing a numerical result

In the original problem, the focus was on indefinite integration, which involves using integration rules such as the Power Rule to obtain a general form solution. Integration is a versatile tool and understanding its concepts deeply enables you to solve a variety of complex problems across different fields.