Problem 64 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

\( \frac{x^4}{4} + \frac{5x^3}{3} + 6x + C \).

Step 1: Break Down the Integral

The integral we need to solve is \( \int (x^3 + 5x^2 + 6) \, dx \). This expression can be broken down into three separate integrals: \( \int x^3 \, dx \), \( \int 5x^2 \, dx \), and \( \int 6 \, dx \). We will solve each of these separately and then combine the results.

Step 2: Integrate \( x^3 \)

To integrate \( x^3 \), use the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Thus, \( \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} \). Since this is an indefinite integral, we include the constant of integration, \( C \).

Step 3: Integrate \( 5x^2 \)

Using the constant multiple rule and power rule, \( \int 5x^2 \, dx = 5 \int x^2 \, dx = 5 \times \frac{x^{2+1}}{2+1} = 5 \times \frac{x^3}{3} = \frac{5x^3}{3} \). Remember to include the constant of integration \( C \), but we'll combine these constants later.

Step 4: Integrate Constant 6

For the constant 6, the integral \( \int 6 \, dx \) results in \( 6x \), as the integral of a constant \( a \) is \( ax + C \). Don't forget the constant of integration, \( C \).

Step 5: Combine the Results

Combine the results of all separate integrals. Therefore, \( \int (x^3 + 5x^2 + 6) \, dx = \frac{x^4}{4} + \frac{5x^3}{3} + 6x + C \). Here, we include a single constant of integration \( C \), combining the constants from each part.

Key Concepts

Integration Techniques

The process of integration involves finding the integral of a function, which represents the area under its curve on a graph. In our exercise, we deal with an indefinite integral, which doesn't have set boundaries and includes a constant of integration. There are several integration techniques, each suitable for different types of functions:

• Power Rule: This is used when the function is a polynomial. It's efficient and straightforward, as you see in this exercise.
• Constant Multiple Rule: If a function is multiplied by a constant, we can take that constant outside the integral and multiply it with the result after integrating.
• Integration by Parts: This technique is more advanced, often used for products of functions.
• Substitution Rule: Used when dealing with composite functions by changing variables to simplify the integral.

In most cases, combining different methods leads to an easier calculation process. In this exercise, we primarily use the power rule and constant multiple rule to solve the indefinite integral.

Power Rule for Integration

The power rule for integration is both simple and crucial in handling polynomials. It's a cornerstone of integration techniques because it applies directly whenever you have terms like \( x^n \). The rule is: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n eq -1 \). This formula allows you to quickly find the antiderivative of \( x^n \), turning differentiation backward. In the given problem, each term of the polynomial \( x^3 + 5x^2 + 6 \) is tackled individually using the power rule:

• For \( x^3 \), applying the power rule gives \( \frac{x^4}{4} + C \).
• For \( 5x^2 \), we use the constant multiple rule along with the power rule to get \( 5 \cdot \frac{x^3}{3} \), simplifying to \( \frac{5x^3}{3} + C \).
• The constant 6 integrates to \( 6x + C \).

Each segment contributes its own form of the power rule, and we combine the results, mindful of \( C \) for the whole integral.

Constant of Integration

When dealing with indefinite integrals, the constant of integration \( C \) plays an essential role and symbolizes an entire family of functions. In an indefinite integral:\[ \int f(x) \, dx = F(x) + C \]\( C \) represents any constant value. Since differentiation eliminates constants, adding \( C \) back distinguishes these antiderivatives when reversing the process. Let's break down its importance:

• Without \( C \), two functions differing only by a constant could appear identical.
• Including \( C \) captures every potential solution of an indefinite integral, reflecting the true range of antiderivatives.
• When combining terms, like in our exercise \( \int (x^3 + 5x^2 + 6) \, dx \), the separate constants of integration from each term are combined at the end as a single \( C \), simplifying representation.

Understanding \( C \) ensures accurate solution of indefinite integrals and conveys the complete solution to those studying areas under curves, even without defined boundaries.