Problem 12 - Calculus and its applications

Question

Find each integral. $$ \int \frac{1}{x^{5}} d x $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{4}x^{-4} + C \).

Step 1: Identify the Form

Recognize that the integral \( \int \frac{1}{x^5} \, dx \) can be rewritten using exponent notation as \( \int x^{-5} \, dx \). This allows us to apply the power rule for integration.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -5 \), so apply the rule: \( \int x^{-5} \, dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C \).

Step 3: Simplify the Expression

Simplify the result from the power rule: \( \frac{x^{-4}}{-4} + C = -\frac{x^{-4}}{4} + C \). This can be further rewritten as \( -\frac{1}{4}x^4 + C \).

Step 4: Verify the Solution

Differentiate the result to ensure it matches the original integrand. Compute \( \frac{d}{dx}\left( -\frac{1}{4} x^{-4} + C \right) \). Using the power rule of differentiation, \( \frac{d}{dx}\left( x^n \right) = n x^{n-1} \), gives \( -\frac{1}{4}( -4 x^{-5}) = \frac{1}{x^{5}} \), confirming the integral is correct.

Key Concepts

Integration

Integration is a fundamental concept in calculus that works as a counterpart to differentiation. When integrating a function, you're essentially finding the area under its curve across a specified range. This area is important in many fields such as physics and engineering for solving real-life problems.

• Integration can take two forms: definite and indefinite. When you don't have specified limits for integration, it's called an indefinite integral, and the result includes a constant of integration, denoted as \( C \).
• A definite integral, on the other hand, calculates the area between the curve and the x-axis over a particular interval.
• To compute an integral, you often use integration formulas and rules, like the power rule, substitution, and integration by parts.

In our exercise, we need to find the indefinite integral of a function. By rewriting \( \frac{1}{x^5} \) as \( x^{-5} \), we set the stage to use the power rule successfully.

Power Rule

The power rule is a handy tool in both integration and differentiation. For integration, it provides a straightforward method to evaluate functions of the form \( x^n \).

• The power rule for integration states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), given \( n eq -1 \).
• This rule is quite useful for any polynomial or simple power expression, even those with negative exponents.

In the exercise, we apply the power rule to \( x^{-5} \). By integrating using this rule, we increment the exponent by one, getting \( \frac{x^{-4}}{-4} \), which is then simplified to \( -\frac{1}{4}x^{-4} + C \). Using the power rule simplifies many integration tasks, making them much more manageable.

Definite Integral

Although the current exercise deals with an indefinite integral, it's important to understand the concept of a definite integral.

• A definite integral not only involves an antiderivative but also allows you to find the exact numerical value representing the area beneath a curve between two points.
• The notation \( \int_{a}^{b} f(x) \, dx \) signifies the integration of \( f(x) \) from \( x = a \) to \( x = b \). This results in a numerical value, different from the indefinite integral which includes \( C \).
• Definite integrals are widely used in practical applications, from calculating distances and volumes to work done and heat transfer.

Understanding how definite integrals relate to areas provides insight into how and why integration is such a valuable tool in calculus.

Antiderivative

An antiderivative is a function whose derivative is a given function. In essence, finding an integral is akin to finding an antiderivative.

• Every function can have multiple antiderivatives, differing by a constant \( C \), which is why we add the constant of integration when finding indefinite integrals.
• In practice, checking your work involves differentiating your antiderivative. If you differentiate correctly and reach the original function, you have successfully determined the antiderivative.
• This verification happens in the exercise when differentiating \( -\frac{1}{4} x^{-4} + C \) yields \( \frac{1}{x^5} \), showing correctness.

Antiderivatives help in solving integrals, and their understanding strengthens your grasp of how integration ties into the broader calculus landscape.