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Chapter 5: Problem 2

Find each indefinite integral. \(\int x^{7} d x\)

Short Answer

Expert verified
\( \int x^7 \, dx = \frac{x^8}{8} + C.\)

Step by step solution

01

Recall the Power Rule for Integration

The Power Rule for Integration states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\) where \(n eq -1\). This is similar to the backward process of differentiation using the Power Rule.
02

Apply the Power Rule

Apply the Power Rule to integrate \(x^7\). Increase the exponent by 1 (which makes it 8), and then divide by the new exponent. Therefore, \[ \int x^7 \,dx = \frac{x^{8}}{8} + C. \]
03

Simplify the Expression

There isn't any complex simplification needed here since there is only a single term. Ensure you've included the constant of integration \(C\), which accounts for any constant that could have been in the original function.

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Key Concepts

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

Power Rule for Integration
The power rule for integration is a handy tool when dealing with polynomials. Essentially, it's like the reverse of differentiating powers of x.
To integrate using this rule, you need to raise the exponent of x by one and then divide by this new exponent.
Here鈥檚 how it works:
  • Let's say you have an integral of the form \( \int x^n \, dx \).
  • First, increase the exponent by 1, turning \( n \) into \( n+1 \).
  • Then, divide the result by the new exponent \( n+1 \).
Putting this all together, the integral is \( \frac{x^{n+1}}{n+1} + C \), where the \( C \) represents the constant of integration. This rule doesn鈥檛 apply when \( n = -1 \) since that would involve dividing by zero, which is mathematically undefined.
Using this simple rule, integration of polynomials becomes a straightforward and methodical process!
Constant of Integration
The constant of integration, often denoted as \( C \), is an essential part of indefinite integrals. It represents the family of functions that differ by a constant.Here's why it's important:
  • When we find integrals, we鈥檙e essentially reversing the differentiation process.
  • The derivative of any constant is zero, so when integrating, we must add an unknown constant that could have been lost previously.
Without this constant, you might miss an infinite number of possibilities that the original function could have had.
It鈥檚 like remembering to mention every possible starting point of a journey when only the steps are known. Always include \( C \) to signify that a solution is not just a single function but a whole family of outcomes!
Exponent Manipulation
Manipulating exponents is key when applying the power rule for integration. Understanding exponent rules is essential for integrating accurately.Let's break it down:
  • When you increase an exponent, you鈥檙e essentially adding one to it; e.g., from \( x^7 \) to \( x^8 \).
  • The manipulation allows the integration process to create a new term whose derivative matches the original one.
Each time you adjust an exponent in the power rule, it helps find an indefinite integral solution.
Remember, exponent manipulation is central not just in integration but throughout algebra, and having strong skills in it will aid in solving integrals seamlessly!

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Most popular questions from this chapter

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{2}^{3} \frac{x^{2}}{x^{3}-7} d x $$

Suppose that you have a positive function and you approximate the area under it using Riemann sums with midpoint rectangles. Explain why, if the function is linear, you will always get the exact area, no matter how many (or few) rectangles you use. [Hint: Make a sketch.]

Suppose that a company found its sales rate (in sales per day) if it did advertise, and also its (lower) sales rate if it did not advertise. If you integrated "upper minus lower" over a month, describe the meaning of the number that you would find.

After \(t\) hours of work, a bank clerk can process checks at the rate of \(r(t)\) checks per hour for the function \(r(t)\) given below. How many checks will the clerk process during the first three hours (time 0 to time 3 )? $$ r(t)=-t^{2}+90 t+5 $$

Find a formula for \(\int_{a} c d x .\) [Hint: No calculation necessary-just think of a graph.]
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