Problem 48 - Applied Calculus

Question

Find the indefinite integrals. $$ \int\left(x^{3}+5 x^{2}+6\right) d x $$

Step-by-Step Solution

Answer

\( \int \left( x^3 + 5x^2 + 6 \right) \, dx = \frac{x^4}{4} + \frac{5x^3}{3} + 6x + C \)

Step 1: Understanding the Problem

We need to find the indefinite integral \( \int \left( x^3 + 5x^2 + 6 \right) \, dx \). This means we need to find an antiderivative for the polynomial function.

Step 2: Recall the Power Rule for Integration

The power rule for integration states that the indefinite integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

Step 3: Apply the Power Rule to Each Term

Integrate each term of the polynomial separately: \[ \int x^3 \, dx = \frac{x^{4}}{4} + C_1 \]\[ \int 5x^2 \, dx = 5 \cdot \frac{x^{3}}{3} + C_2 \]\[ \int 6 \, dx = 6x + C_3 \]

Step 4: Combine the Integrated Terms

Combine the results of integrating each term.The antiderivative is:\[ \frac{x^4}{4} + \frac{5x^3}{3} + 6x + C \]where \( C \) is the constant of integration which accumulates all previous constants.

Key Concepts

Polynomial Integration

Polynomial integration is the process of taking the integral of a polynomial function, which is a sum of terms where each term is a constant multiplied by a power of the variable, typically written as \( x \). In the context of calculus, when we deal with polynomials, finding the integral helps us determine the area under the curve represented by the polynomial on a graph. This is especially useful in understanding accumulated quantities such as distance, area, and other physical concepts where the value changes continuously.

• Polynomials are structured as sums of powers of variables, such as \( x^3 + 5x^2 + 6 \).
• Integration reverses differentiation, allowing us to find antiderivatives or functions whose derivative gives us our original polynomial function.
• Indefinite integrals, unlike definite integrals, do not have limits of integration, resulting in a general expression that includes a constant of integration, \( C \).

Understanding polynomial integration demystifies a major component of calculus, making it easier to tackle practical and theoretical problems.

Power Rule for Integration

The power rule for integration is a fundamental technique used to integrate expressions where a variable is raised to a power. This rule simplifies obtaining an antiderivative by applying a straightforward formula. For any term \( x^n \), the integral is calculated as:
\[\frac{x^{n+1}}{n+1} + C\]
Where \( n \) is not equal to \(-1\) and \( C \) represents the constant of integration. The constant shows that we could shift the antiderivative vertically, and it would still yield the same derivative.

• Each term in a polynomial is integrated separately using the power rule.
• The constant factor of a term, such as the "5" in \( 5x^2 \), is retained outside the integration process and multiplied after applying the power rule.
• This rule dramatically simplifies the integration process, especially in larger polynomials or complex functions.

Mastering the power rule provides a strong foundation for integrating more complex functions later in calculus studies.

Antiderivative Calculation

Calculating the antiderivative, also known as integration, involves finding a function that, when differentiated, results in the original function. This step-by-step process transforms a given polynomial into its antiderivative using integration rules.

• First, apply the power rule individually to each term of the polynomial.
• Gather all the integrated terms together. If multiple terms exist, like in our original expression \( x^3 + 5x^2 + 6 \), integrate them separately first, then combine the results.
• Don't forget the constant of integration, \( C \), which acknowledges that many functions share the same derivative but differ by a constant.

After integration, the expression \( \frac{x^4}{4} + \frac{5x^3}{3} + 6x + C \) provides a comprehensive antiderivative. By mastering these basic calculations, you prepare yourself for more advanced applications in calculus and its various fields.