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Chapter 25: Problem 30

Find antiderivatives of the given functions. $$f(R)=3\left(R^{2}+1\right)^{2}(2 R)$$

Short Answer

Expert verified
The antiderivative is \( F(R) = R^6 + 3R^4 + 3R^2 + C \).

Step by step solution

01

Expand the given function

The given function is \( f(R) = 3(2R)(R^2 + 1)^2 \). Start by expanding \((R^2 + 1)^2\) using the binomial theorem: \[(R^2 + 1)^2 = (R^2)^2 + 2(R^2)(1) + 1^2 = R^4 + 2R^2 + 1\] Substitute back into the function: \[ f(R) = 3(2R)(R^4 + 2R^2 + 1) \] Simplify further: \[ f(R) = 3(2R)(R^4) + 3(2R)(2R^2) + 3(2R)(1) \] This simplifies to: \[ f(R) = 6R^5 + 12R^3 + 6R \]
02

Find the antiderivative of each term

To find the antiderivative of \( f(R) = 6R^5 + 12R^3 + 6R \), integrate each term separately. Recall the power rule for integration: \( \int R^n \, dR = \frac{R^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.- **For \( 6R^5 \):** \[ \int 6R^5 \, dR = 6 \cdot \frac{R^{6}}{6} = R^6 \]- **For \( 12R^3 \):** \[ \int 12R^3 \, dR = 12 \cdot \frac{R^{4}}{4} = 3R^4 \]- **For \( 6R \):** \[ \int 6R \, dR = 6 \cdot \frac{R^{2}}{2} = 3R^2 \]
03

Combine the antiderivatives

Combine all the antiderivative terms to form the general antiderivative of the function:\[ F(R) = R^6 + 3R^4 + 3R^2 + C \]where \( C \) is the constant of integration.

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful tool used to expand expressions that are raised to a power. For an expression of the form \((a+b)^n\), the theorem provides a way to expand this into a sum of terms involving coefficients, powers of \(a\), and powers of \(b\).
The expression \((R^2 + 1)^2\) can be expanded using the Binomial Theorem as follows:
  • First, square each term: \((R^2)^2 + 2(R^2)(1) + 1^2\).
  • This results in \(R^4 + 2R^2 + 1\).
This expansion simplifies the process of finding antiderivatives later, as it breaks the expression into simple terms.
Power Rule for Integration
The Power Rule for Integration is a fundamental concept in calculus that is used to find antiderivatives of power functions. It states that if you want to integrate a term like \(R^n\), the result is \(\frac{R^{n+1}}{n+1} + C\), where \(C\) is a constant of integration.
When applying this rule, keep the following in mind:
  • The exponent \(n\) is increased by one.
  • The term is divided by the new exponent.
For example:
  • The integral of \(6R^5\) becomes \(R^6\) after applying the power rule.
  • The integral of \(12R^3\) results in \(3R^4\).
  • The integral of \(6R\) simplifies to \(3R^2\).
Adding these results together gives the complete antiderivative of the function.
Polynomial Functions
Polynomial functions are expressions that consist of variables raised to non-negative integer exponents and have constant coefficients. These functions can be as simple as \(R\) or more complex like \(R^5 + 2R^2 + 4\).
Key characteristics include:
  • Each term is called a 'monomial.'
  • The highest power of the variable is the 'degree' of the polynomial.
  • Polynomials are continuous and smooth curves when graphed.
Understanding polynomial functions is essential as they frequently appear in calculus problems, requiring differentiation or integration. For the function \(6R^5 + 12R^3 + 6R\), each term is a polynomial, and finding their antiderivative involves integrating each term separately using the power rule.

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Most popular questions from this chapter

Find the exact area under the given curves between the indicated values of \(x\). The functions are the same as those for which approximate areas were found in Exercises \(5-14\). \(y=2 x+1,\) between \(x=0\) and \(x=2\)

Find the approximate area under the curves of the given equations by dividing the indicated intervals into n subintervals and then add up the areas of the inscribed rectangles. There are two values of n for each exercise and therefore two approximations for each area. The height of each rectangle may be found by evaluating the function for the proper value of \(x\). See Example 1. \(y=3 x,\) between \(x=0\) and \(x=3,\) for (a) \(n=3(\Delta x=1),\) (b) \(n=10(\Delta x=0.3)\)

Solve each given problem by using the trapezoidal rule. The rate \(\frac{d A}{d t}\) (in standard pollution index per hour) of a pollutant put into the air by a smokestack is given by \(\frac{d A}{d t}=\frac{150}{1+0.25(t-4.0)^{2}}+25,\) where \(t\) is the time (in \(\mathrm{h}\) ) after 6 A.M. With \(n=6,\) estimate the total amount of the pollutant put into the air between 6 A.M. and noon.

Find the approximate area under the curves of the given equations by dividing the indicated intervals into n subintervals and then add up the areas of the inscribed rectangles. There are two values of n for each exercise and therefore two approximations for each area. The height of each rectangle may be found by evaluating the function for the proper value of \(x\). See Example 1. \(y=\frac{1}{\sqrt{x+1}},\) between \(x=3\) and \(x=8,\) for (a) \(n=5,\) (b) \(n=10\)

Find the exact area under the given curves between the indicated values of \(x\). The functions are the same as those for which approximate areas were found in Exercises \(5-14\). \(y=\frac{1}{\sqrt{x+1}},\) between \(x=3\) and \(x=8\)
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