Chegg Calculate T6 for The Following Integral X 3 Dx

This guide explains how to calculate the integral of x³ dx using Chegg's T6 method. We'll cover the formula, step-by-step calculation, and provide a working calculator to compute the result quickly.

Introduction

Calculating the integral of x³ dx is a fundamental operation in calculus. The integral of x³ with respect to x is a basic power rule application. This guide will walk you through the process using Chegg's T6 method, which provides a clear and systematic approach to solving integrals.

The integral of x³ dx represents the area under the curve of the function x³. This concept is essential in physics, engineering, and economics for finding areas, volumes, and other quantities that involve accumulation.

Formula

The basic formula for integrating x³ is derived from the power rule of integration:

∫x³ dx = (x⁴)/4 + C

Where C is the constant of integration.

This formula tells us that the integral of x³ is x⁴ divided by 4, plus an arbitrary constant C. The constant C accounts for any initial conditions that might be present in a specific problem.

Calculation

To calculate the integral of x³ dx using Chegg's T6 method, follow these steps:

Identify the integrand: In this case, the integrand is x³.
Apply the power rule: For any term of the form xⁿ, the integral is (xⁿ⁺¹)/(n+1).
Add the constant of integration: Since we're dealing with an indefinite integral, we must include the constant C.

Let's apply these steps to the integral of x³ dx:

The integrand is x³, which can be written as x³.
Using the power rule, we have ∫x³ dx = (x⁴)/4.
Adding the constant of integration gives us the final result: (x⁴)/4 + C.

This step-by-step approach ensures that you understand how to apply the power rule to solve integrals of the form x³ dx.

Example

Let's work through a concrete example to illustrate how to calculate the integral of x³ dx using Chegg's T6 method.

Suppose we want to find the integral of x³ from x = 0 to x = 2. Here's how we would approach it:

First, find the antiderivative of x³, which we know is (x⁴)/4 + C.
Evaluate the antiderivative at the upper limit (x = 2): (2⁴)/4 = 16/4 = 4.
Evaluate the antiderivative at the lower limit (x = 0): (0⁴)/4 = 0.
Subtract the lower limit evaluation from the upper limit evaluation: 4 - 0 = 4.

The result is 4, which represents the area under the curve of x³ from x = 0 to x = 2.

Note: The constant of integration C cancels out when evaluating definite integrals, so it doesn't affect the final result in this case.

FAQ

What is the integral of x³ dx?

The integral of x³ dx is (x⁴)/4 + C, where C is the constant of integration. This is derived from the power rule of integration.

Why do we add the constant of integration C?

The constant of integration C is added because indefinite integrals represent a family of functions that differ by a constant. It accounts for any initial conditions that might be present in a specific problem.

How do I evaluate a definite integral of x³?

To evaluate a definite integral of x³, first find the antiderivative (x⁴)/4 + C. Then evaluate the antiderivative at the upper and lower limits, subtract the lower limit evaluation from the upper limit evaluation, and the result is the area under the curve between those limits.

What is the difference between an indefinite and definite integral?

An indefinite integral represents a family of functions that differ by a constant, while a definite integral represents a specific numerical value. Definite integrals have upper and lower limits, while indefinite integrals do not.