Problem 17 - Calculus and its applications

Question

Find each integral. $$ \int \frac{d x}{x^{4}} $$

Step-by-Step Solution

Answer

\( -\frac{1}{3x^{3}} + C \)

Step 1: Identify the integral form

The given integral is \( \int \frac{d x}{x^{4}} \). This can be rewritten using the negative exponent form as \( \int x^{-4} \, dx \).

Step 2: Use the Power Rule for Integration

The integral \( \int x^{n} \, dx \) can be solved using the power rule, which states: \( \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -4 \).

Step 3: Apply the Power Rule

Substituting \( n = -4 \) into the formula gives: \( \frac{x^{-4+1}}{-4+1} + C \). Simplifying yields \( \frac{x^{-3}}{-3} + C \).

Step 4: Simplify the expression

The expression \( \frac{x^{-3}}{-3} + C \) simplifies to \( -\frac{1}{3}x^{-3} + C \).

Step 5: Convert back to positive exponent form

Convert \( x^{-3} \) to the positive exponent form, yielding \( -\frac{1}{3} \cdot \frac{1}{x^{3}} + C \), or equivalently, \( -\frac{1}{3x^{3}} + C \).

Key Concepts

Power Rule for Integration

The Power Rule is a fundamental tool in calculus for finding the antiderivative of a function. It's particularly useful when dealing with polynomials or expressions that can be turned into polynomials. The rule states that if you have an integral of the form \( \int x^n \, dx \), you can integrate it using the formula:
\[ \frac{x^{n+1}}{n+1} + C \]
where \( n \) is any real number except \(-1\), and \( C \) represents the constant of integration. It’s important to remember:

• \( n eq -1 \) because this would involve dividing by zero.
• The constant \( C \) is crucial because indefinite integrals represent a family of functions, not just one solution.

To apply this rule, simply increase the exponent by one and divide by the new exponent. Let's take an example:
For \( n = -4 \), you would rewrite the integral \( \int x^{-4} \, dx \) using:
\[ \frac{x^{-4+1}}{-4+1} + C = \frac{x^{-3}}{-3} + C \]
This clearly shows how the power rule simplifies the integration process dramatically.

Negative Exponent

Expressions with negative exponents can sometimes seem a bit tricky, but they follow a specific and simple rule. A negative exponent indicates that the base, in our case \( x \), is on the denominator of a fraction:
\[ x^{-n} = \frac{1}{x^n} \]
This means that if you see a term like \( x^{-4} \), it is equivalent to \( \frac{1}{x^4} \). Integrals can sometimes be simpler to evaluate if we write terms with negative exponents as fractions instead:

• This helps in visualization, showing the term is part of the divisor.
• Transitioning between exponent forms can also simplify the integration process overall.

In our exercise, rewriting \( \frac{d x}{x^4} \) as \( x^{-4} \, dx \) aligns it with the usual power rule for integration format, making it easier to apply the rule.

Simplifying Integrals

Simplifying integrals is an important step in the process of integration, as it helps clarify and correct the solution at each stage. After finding the antiderivative, the resulting expression can often be further simplified for clarity and practicality:
1. **Combining Terms**: Always look to combine like terms, if possible, to reduce the complexity of the expression.
2. **Exponent Conversion**: Convert any negative exponents back to a positive form if desired. For example, changing \( x^{-3} \) to \( \frac{1}{x^3} \).
In our solution, the integral \( \int \frac{d x}{x^4} \) is simplified step-by-step:

• Initially expressed in terms of negative exponents as \( \int x^{-4} \, dx \).
• After applying the power rule, it was simplified to \( -\frac{1}{3}x^{-3} + C \).
• Finally, it was converted to the positive exponent format, \( -\frac{1}{3x^3} + C \), which is often a more standardized way to express the result.

This multi-step simplification clarifies the outcome and ensures that the answer is easily interpretable.