Problem 42 - Applied Calculus

Question

Find the indefinite integrals. $$ \int\left(x^{5}-12 x^{3}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{x^6}{6} - 3x^4 + C \).

Step 1: Distribute the Integral

First, distribute the integral across the terms inside the integrand. This involves applying the integral separately to each term. Our expression becomes:\[\int x^{5} dx - \int 12x^{3} dx\]

Step 2: Apply the Power Rule for Integration

Integrate each term individually using the power rule. Recall, the power rule for integration is \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.For \( \int x^5 dx \):\[\frac{x^{5+1}}{5+1} = \frac{x^6}{6}\]For \( \int 12x^3 dx \):\[12 \times \frac{x^{3+1}}{3+1} = 12 \times \frac{x^4}{4} = 3x^4\]

Step 3: Combine the Results

Combine the integrated results along with the constant of integration C:\[\frac{x^6}{6} - 3x^4 + C\]This is the indefinite integral of the given function. Here, \( C \) is an arbitrary constant.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental concept that simplifies the process of finding indefinite integrals of polynomial functions. This rule states that if you have a function in the form of a power of x, such as \( x^n \), its integral can be found using the formula:

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

This means you increase the exponent by one and then divide by the new exponent. The constant \( C \) represents the constant of integration, which you'll learn more about shortly.

When using the Power Rule, it's important first to identify if each term in your integral can be expressed as \( x^n \). This only applies to terms that are powers of x, so it's crucial to simplify complex expressions first. If you encounter a term like \( 3x^n \), simply multiply the result from the Power Rule by the constant coefficient (here, 3). It's all about applying this rule to each term individually and then summing the results.

Constant of Integration

Understanding the constant of integration, denoted as \( C \), is essential when working with indefinite integrals. Since indefinite integrals aim to find antiderivatives, there are infinitely many possible solutions that differ by just a constant. This is because the derivative of a constant is zero.

When you integrate a function, adding \( C \) accounts for these potential variations. For example, the integral of \( \int x^2 \, dx \) is \( \frac{x^3}{3} + C \). If your original function could have been \( f(x) = \frac{x^3}{3} + 7 \), \( C \) would account for that 7.

It's critical to include \( C \) in your final answer when solving indefinite integrals. Without it, your solution may be incomplete or incorrect. Always remember that the constant of integration reflects the unknown constant that could be subtracted during differentiation.

Integration Techniques

When integrating complex functions, several techniques may be employed. Distributing the integral across terms, as shown in this example, is one straightforward but powerful technique. Here’s how you can make integration more manageable:

• **Distribute:** Break down a polynomial into separate terms, allowing you to integrate each term individually. This is highly beneficial for expressions like \( \int (x^5 - 12x^3) \, dx \).
• **Substitution:** Useful when an integral involves a composite function or requires changing variables to simplify it.
• **Integration by Parts:** This technique is for products of functions, derived from the product rule for differentiation.

Each method has its use, depending on the structure of the integrand. Practicing different techniques will improve your integration skills, helping you tackle more challenging problems with confidence.