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Chapter 6: Problem 6

Find the antiderivatives of the functions: $$ (5 x+1)^{2} $$

Short Answer

Expert verified
Antiderivative: \(\frac{25}{3}x^3 + 5x^2 + x + C\).

Step by step solution

01

Expand the Expression

First, we expand the expression \((5x + 1)^2\) using the formula for the square of a binomial:\[(a + b)^2 = a^2 + 2ab + b^2\]For \((5x + 1)^2\), let \(a = 5x\) and \(b = 1\). So, the expanded form is:\[(5x)^2 + 2(5x)(1) + 1^2 = 25x^2 + 10x + 1\].
02

Find the Antiderivative Term-by-Term

Next, we find the antiderivative of each term in the polynomial \(25x^2 + 10x + 1\):1. The antiderivative of \(25x^2\) is \(\frac{25}{3}x^3\).2. The antiderivative of \(10x\) is \(\frac{10}{2}x^2 = 5x^2\).3. The antiderivative of \(1\) is \(x\).Don't forget to include the constant of integration, \(C\).
03

Combine the Antiderivatives

Combine the antiderivatives of each term to get the overall antiderivative of the expression. Thus:\[\frac{25}{3}x^3 + 5x^2 + x + C\].

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Key Concepts

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

Binomial Expansion
The binomial expansion is a crucial concept whenever you encounter expressions like \((5x + 1)^2\). To handle this, we often use the binomial theorem, which helps us expand expressions raised to a power.

To expand a binomial of the form \((a + b)^2\), you apply the identity:
  • \((a + b)^2 = a^2 + 2ab + b^2\).
For instance, consider our expression \((5x + 1)^2\). Here, \(a = 5x\) and \(b = 1\). Using the formula, the expansion becomes:
  • \((5x)^2 + 2 \times (5x) \times 1 + 1^2\)
  • This results in: \(25x^2 + 10x + 1\).
By expanding the binomial, we transform a potentially complex expression into a polynomial, paving the way for easier integration.
Polynomial Integration
Polynomial integration is the next step after expanding your expression. It involves finding the antiderivatives of polynomial terms.

For each term in the polynomial, you apply the power rule of integration:
  • If you have a term \(ax^n\), its antiderivative is \(\frac{a}{n+1}x^{n+1}\).
For the polynomial \(25x^2 + 10x + 1\):
  • The antiderivative of \(25x^2\) is \(\frac{25}{3}x^3\).
  • For \(10x\), it becomes \(5x^2\) because \(\frac{10}{2}x^2 = 5x^2\).
  • The constant \(1\) integrates to \(x\) since its antiderivative is \(x^1\).
Once you find each component's antiderivative, combine them, and remember to include an integration constant, forming the overall integrated expression.
Constant of Integration
When performing integration, especially when dealing with indefinite integrals, we always add a constant of integration, denoted as \(C\).

This constant appears because integration represents the family of all functions whose derivative is the integrand.
Since the derivative of a constant is zero, adding a constant does not affect differentiation results, but is crucial for completeness:
  • The antiderivative of a number, say \(1\), integrates to \(x + C\).
  • Therefore, whatever the function's original constant was, the integration process accounts for it by using \(C\).
By including this constant, you ensure that your solution encompasses all possible functions meeting the original conditions of the equation.

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