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Chapter 4: Problem 12

Determine these indefinite integrals. $$\int \frac{1}{x^{5}} d x$$

Short Answer

Expert verified
-\frac{1}{4x^4} + C

Step by step solution

01

Identify the integral form

Recognize that the integral given is \(\int \frac{1}{x^{5}} dx\). This can be rewritten by expressing the denominator with a negative exponent.
02

Rewrite the integrand

Rewrite \(\frac{1}{x^{5}}\) as \(x^{-5}\). Now the integral becomes \(\int x^{-5} dx\).
03

Apply the power rule for integration

The power rule for integration states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\). Here, \(n = -5\).
04

Integrate using the power rule

Apply the power rule: \(\int x^{-5} dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C = -\frac{1}{4} x^{-4} + C\).
05

Simplify and write the final answer

Rewrite the expression in a simplified form: \(-\frac{1}{4} x^{-4} + C = -\frac{1}{4} \frac{1}{x^4} + C = -\frac{1}{4x^4} + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Power Rule
The power rule for integration is a basic yet essential tool. It helps us find indefinite integrals easily. The rule states:
\(\text{\frac{x^{n+1}}{n+1}}\) + C, where (n 鈮� -1). \br Here, C is the constant of integration.\br By knowing this rule, you turn complex integrals into simpler problems. When you identify the power of x in an integrand, use this rule to integrate it. This makes solving the integral much more straightforward.
Negative Exponents
Negative exponents turn division into multiplication. For example, \(\frac{1}{x^5}\ becomes x^{-5}\). This makes integrals easier to work with. Using negative exponents is often the first step in solving integrals like \(\text{\frac{1}{x^5}}\). Once rewritten as \(x^{-5}\), applying the power rule is simple.\br Negative exponents help simplify expressions, making complex integral problems more manageable. Remember, the trick is to identify when to rewrite fractions using negative exponents.
Integrand Rewriting
Rewriting the integrand is an important step in simplifying integrals. In this task, \(\int \frac{1}{x^5} \), the integrand \(\frac{1}{x^5}\is simplified to x^{-5}\). This makes applying the power rule straightforward.\br By rewriting, you put the integral in a form where rules like the power rule can be easily applied. Look for opportunities to rewrite complex fractions or expressions to simpler forms. This not only simplifies the process, but it also reduces the chance of making errors during integration.

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Most popular questions from this chapter

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