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Chapter 6: Problem 15

Find an antiderivative. $$f(t)=\frac{t^{2}+1}{t}$$

Short Answer

Expert verified
The antiderivative is \( F(t) = \frac{t^2}{2} + \ln|t| + C \).

Step by step solution

01

Simplify the Expression

Before finding the antiderivative, simplify the given function by dividing each term in the numerator by the denominator:\[ f(t) = \frac{t^2 + 1}{t} = \frac{t^2}{t} + \frac{1}{t} = t + \frac{1}{t}. \]
02

Find the Antiderivative of Each Term

Now we find the antiderivative of each term separately. For the term \(t\), the antiderivative is \(\frac{t^2}{2}\), and for \(\frac{1}{t}\), the antiderivative is \(\ln|t|\). Thus, the antiderivative of \(f(t)\) is:\[ F(t) = \frac{t^2}{2} + \ln|t| + C, \]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.

Simplifying Expressions
Before tackling the task of finding an antiderivative, we first need to simplify the given expression. This process can make your life so much easier! For our function \( f(t)=\frac{t^{2}+1}{t} \), simplification involves breaking down terms to their simplest forms by performing algebraic operations. Why Simplify?
  • It makes integration manageable by reducing the function to basic terms.
  • Helps avoid complex calculations later on.
  • Provides a clearer understanding of the behavior of the function.
In our context, the expression \( \frac{t^{2}+1}{t} \) is simplified to \( t + \frac{1}{t} \). This is achieved by splitting the fraction and dividing each term in the numerator by the denominator separately. By simplifying, we can focus on each part individually during integration.
Integration Techniques
Once we've simplified the expression, the next step is to find its antiderivative鈥攂asically the function whose derivative would give us the original function. This requires us to apply certain integration techniques. Let鈥檚 break it down!Finding the Antiderivative
  • For a term like \( t \): It is straightforward to integrate. Using the power rule for integration, which involves increasing the power by 1 and dividing by the new power, we get \( \frac{t^2}{2} \).
  • For \( \frac{1}{t} \): This term is known for having a unique antiderivative, which is \( \ln|t| \). This results from the integral property that \( \int \frac{1}{t} \, dt = \ln|t| \).
Integrating term by term significantly simplifies solving for the antiderivative of composite functions. Once each term is integrated, they are combined to find the complete antiderivative of the function.
Constant of Integration
After finding the antiderivative, don鈥檛 forget to add a super important part: the constant of integration, denoted as \( C \). But why do we do this?Understanding the Constant of Integration
  • When you integrate a function, you are essentially 'undoing' differentiation. However, derivatives of constant values are zero, meaning they disappear during differentiation.
  • This means any number of constant values could have been originally present in the function before differentiation.
Thus, adding \( C \) accounts for all these possible constant values. In our antiderivative \( F(t) = \frac{t^2}{2} + \ln|t| + C \), \( C \) represents an unknown constant that ensures all potential original functions are considered, rendering our solution complete and comprehensive.

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