Problem 35 - Applied Calculus

Question

Find the indefinite integrals. $$ \int 6 x^{2} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

The problem asks to find the indefinite integral of the function \( 6x^2 \) with respect to \( x \). This means you are to determine the antiderivative of the given function.

Step 2: Use the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). In this function, \( x^2 \) fits this rule, where \( n = 2 \), and you must also include the coefficient 6 in your integration process.

Step 3: Apply the Power Rule

Applying the power rule, \( \int 6x^2 \, dx = 6 \cdot \frac{x^{2+1}}{2+1} + C \). Simplifying the expression inside gives \( 6 \cdot \frac{x^3}{3} + C \).

Step 4: Simplify the Expression

Simplify the expression by performing the multiplication. This results in \( 2x^3 + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental technique used to find the antiderivative of functions in the form of a power of a variable. Consider a function, where the variable is raised to a power, such as \( x^n \). The Power Rule allows us to integrate this quickly and effortlessly. Here's how it works.

If you have an integral of the form \( \int x^n \, dx \), you apply the rule:

• Add 1 to the exponent \( n \), resulting in \( n + 1 \).
• Divide by the new exponent \( n+1 \).

Thus, the integral becomes \( \frac{x^{n+1}}{n+1} + C \), where \( C \) represents the constant of integration. This rule applies when \( n eq -1 \).

This method simplifies the process of finding integrals, especially if you are dealing with polynomials or integer exponents. In our exercise, using the Power Rule on \( 6x^2 \) gave us \( 6 \cdot \frac{x^3}{3} \), which simplified to \( 2x^3 \). This process allows us to find the antiderivative in a straightforward manner, which leads us to the next concept.

Antiderivative

An antiderivative, often referred to as an indefinite integral, is a function that reverses the process of differentiation. It gives us the original function (or at least one of many possibilities) before it was differentiated.

When given a function, the antiderivative helps to find out what the function was before any derivatives were applied. For example, in our problem, we sought the antiderivative of \( 6x^2 \). By using the Power Rule, we identified that \( 2x^3 \) functions as an antiderivative. This means that taking the derivative of \( 2x^3 \) would yield the original function, \( 6x^2 \).

Understanding antiderivatives is crucial in calculus. It aids in solving differential equations, finding areas under curves, and is a building block for much more complex calculus problems. Remember that there can be multiple antiderivatives for a single function, which brings us to the concept of constants.

Constant of Integration

When finding an indefinite integral, such as in our example problem, it's important to include the constant of integration. This is usually denoted by \( C \).

But why do we include \( C \)? Consider this: the derivative of any constant is zero. Therefore, when you differentiate an antiderivative with a constant, you lose the constant since differentiating a constant results in zero. Thus, while integrating, we include \( C \) to account for any possible constant that was "lost" during differentiation.

• Every indefinite integral must include +C at the end.
• It represents a family of functions differing by a constant.

By including \( C \) in \( 2x^3 + C \), we acknowledge that \( 2x^3 + 5 \), \( 2x^3 - 3 \), etc., are equally valid antiderivatives, differing only by a constant. This signifies the inherent flexibility and generality of indefinite integrals.