Problem 34 - Applied Calculus

Question

Find the indefinite integrals. $$ \int 9 x^{2} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(9x^2\) is \(3x^3 + C\).

Step 1: Identify the Integral Form

We need to find the indefinite integral of the function \(9x^2\). This is a polynomial function, which allows us to use the power rule for integration.

Step 2: Apply the Power Rule

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\). Applying this rule to \(9x^2\), we increase the exponent by 1 and divide by the new exponent.

Step 3: Integrate the Function

Integrate each part of the function separately: \(\int 9x^2 \, dx = 9 \cdot \int x^2 \, dx = 9 \cdot \frac{x^{2+1}}{2+1} + C = 9 \cdot \frac{x^3}{3} + C\). Simplify to get \(3x^3 + C\).

Step 4: Write the Final Answer

The indefinite integral of \(9x^2\) is \(3x^3 + C\), where \(C\) is the constant of integration.

Key Concepts

Power Rule for Integration

Integration is a fundamental concept in calculus, particularly in finding the area under curves. The power rule for integration is an essential tool for integrating polynomial functions.
The general form of the power rule is expressed as:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \(n\) is any real number except \(-1\), and \(C\) represents the constant of integration.

• To use the power rule, increase the power of \(x\) by one.
• Divide by the new power.

For example, when integrating \(9x^2\), increase the exponent from 2 to 3 and divide 9 by 3, yielding \(3x^3 + C\).
This rule simplifies the process of integrating polynomials and is used extensively in both simple and complex mathematical problems.

Polynomial Function

A polynomial function is a mathematical expression consisting of variables raised to various powers and multiplied by coefficients. Examples include \(x^2, 3x^4,\) and \(7x^3 - 4x + 2\). Such functions are crucial in calculus as they are straightforward to manipulate and integrate.
A key feature of polynomial functions is their simple and predictable behavior when integrating, particularly using the power rule.

• The integration of a polynomial function involves applying the power rule to each term individually.
• Polynomial functions are smooth and continuous, which means they don't have sharp corners or jumps.

Understanding polynomial functions helps greatly in solving integration problems, providing a solid foundation for tackling more complex calculus concepts. Recognizing these functions ensures efficient use of integration rules.

Constant of Integration

When finding an indefinite integral, the result includes a term known as the constant of integration, typically denoted as \(C\).
This constant is added because integration is the reverse process of differentiation, and when differentiating, any constant term would vanish.

• The constant of integration reflects that there is an infinite number of antiderivatives for any given function.
• It is a crucial part of indefinite integrals because it completes the family of solutions.

For instance, in the solution \(3x^3 + C\), the \(C\) accounts for any vertical shifting of the function 3x^3.
Being mindful of including \(C\) in your solutions ensures accuracy and comprehensiveness when dealing with indefinite integrals, as it indicates all possible antiderivatives of a function.