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Chapter 4: Problem 5

Determine these indefinite integrals. $$\int x^{1 / 4} d x$$

Short Answer

Expert verified
\( \frac{4}{5} x^{5/4} + C \)

Step by step solution

01

Identify the integral form

The given integral is in the form of a power of x: \ \(\int x^{1/4} \,dx \).
02

Use the power rule for integration

The power rule for integration states that \ \(\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\) for any real number \(n eq -1\). Here, \(n = \frac{1}{4}\).
03

Apply the power rule

Apply the power rule with \(n = \frac{1}{4}\). Therefore, \ \( \int x^{1/4} \,dx = \frac{x^{(1/4) + 1}}{(1/4) + 1} + C = \frac{x^{5/4}}{5/4} + C \).
04

Simplify the result

Simplify the fraction \( \frac{x^{5/4}}{5/4} \) by multiplying the numerator by the reciprocal of the denominator: \ \( \frac{x^{5/4}}{5/4} = \frac{4}{5} x^{5/4} \). Therefore, \ \( \int x^{1/4} \,dx = \frac{4}{5} x^{5/4} + C\).

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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 fundamental tool in calculus. It allows us to easily find the indefinite integral of any function in the form of a power of x.

The rule states:
\[\text{If} \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \text{where } n eq -1.\]

This formula tells us that when integrating a term like x raised to a power, you need to:
  • Add 1 to the power of x.
  • Divide by the new power.
  • Remember to add the constant of integration, C, which accounts for any constant value that could have been present before differentiation.
Using this rule simplifies many problems in calculus and is key to solving integrals of simple polynomial functions.
Integral of x^n
To understand the power rule better, let's focus on the specific case of integrating functions of the form x^n.

For the integral \(\int x^n \, dx\), you're essentially reversing the differentiation process of x^n:

  • Identify the exponent n. For instance, if you have \( x^{1/4} \), then \( n = \frac{1}{4} \).
  • Apply the rule: increase the exponent by 1. Here, \( (1/4) + 1 = 5/4 \).
  • Divide by the new exponent: \( \int x^{1/4} \,dx = \frac{x^{5/4}}{5/4} + C \).
This step-by-step method ensures you execute the process correctly. The added constant C is included because the integral of a constant function is not visible and thus must be accounted for. This constant can vary based on the initial conditions of the problem.
Simplifying Fractions in Integrals
Once you use the power rule, you often end up with a fraction. Simplifying these fractions is crucial for the final answer.

Consider the fraction \( \frac{x^{5/4}}{5/4} \):
  • Understand that dividing by \( \frac{5}{4} \) is the same as multiplying by its reciprocal \( \frac{4}{5} \).
  • Rewrite the expression: \( \frac{x^{5/4}}{5/4} = x^{5/4} \times \frac{4}{5} = \frac{4}{5} x^{5/4} \).
This simplification makes the expression cleaner and easier to understand.

Simplifying fractions after integrating ensures you present your final answer in its simplest and most usable form. For instance, in our exercise:
  • Starting from \( \int x^{1/4} \,dx = \frac{x^{5/4}}{5/4} + C \).
  • We simplify it to \( \frac{4}{5} x^{5/4} + C \).
This step is important for clarity and correctness in your solutions.

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Most popular questions from this chapter

Find the error in each of the following. Explain. $$\begin{aligned} \int_{1}^{2}\left(\ln x-e^{x}\right) d x &=\left[\frac{1}{x}-e^{x}\right]_{1}^{2} \\\&=\left(\frac{1}{2}-e^{2}\right)-\left(1-e^{1}\right) \\\&=e-e^{2}-\frac{1}{2}\end{aligned}$$

Use geometry to evaluate each definite integral. $$\int_{0}^{5} 4 x d x$$

The divorce rate in the United States is approximated by \(D(t)=100,000 e^{0.025 t}\) where \(D(t)\) is the number of divorces occurring at time \(t\) and \(t\) is the number of years measured from \(1900 .\) That is, \(t=0\) corresponds to \(1900, t=98 \frac{9}{365}\) corresponds to January \(9,1998,\) and so on. a) Find the total number of divorces from 1900 to \(2005 .\) Note that this is given by \(\int_{0}^{105} D(t) d t\) b) Find the total number of divorces from 1980 to 2006. Note that this is given by \(\int_{80}^{106} \mathrm{D}(t) d t\)

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