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Chapter 4: Problem 11

Find the most general antiderivative of the function.(Check your answer by differentiation.) \(c(t)=\frac{3}{t^{2}}, t>0\)

Short Answer

Expert verified
The most general antiderivative of \(c(t) = \frac{3}{t^2}\) is \(-\frac{3}{t} + C\).

Step by step solution

01

Identify the function

The given function is \(c(t) = \frac{3}{t^2}\). We need to find its antiderivative.
02

Rewrite the function

Express the function in a simpler form for integration: \(c(t) = 3t^{-2}\).
03

Antiderivative Formula

Recall the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\).
04

Apply the Power Rule

Apply the power rule to \(3t^{-2}\): \(\int 3t^{-2} \, dt = 3 \cdot \frac{t^{-1}}{-1} + C = -3t^{-1} + C\).
05

Simplify the Antiderivative

The antiderivative simplifies to \(-\frac{3}{t} + C\), where \(C\) is the constant of integration.
06

Differentiate to Verify

Differentiate the found antiderivative \(-\frac{3}{t} + C\): \(\frac{d}{dt}(-\frac{3}{t} + C) = \frac{3}{t^2}\). The differentiation gives us the original function, confirming the solution.

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Key Concepts

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

Power Rule for Integration
When finding the antiderivative of a function like \(c(t) = \frac{3}{t^2}\), one of the most useful tools is the Power Rule for Integration. This rule helps us integrate functions in the form \(x^n\). By rewriting the function, we simplify the problem. In this case, we change \(\frac{3}{t^2}\) to \(3t^{-2}\). This allows us to apply the Power Rule: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]Here, \(n\) must not equal \(-1\). We apply this to our function:
  • Identify \(n = -2\) and use the formula.
  • Add \(1\) to the exponent: \(-2 + 1 = -1\).
  • Apply the rule: \(\int 3t^{-2} \, dt = 3\cdot \frac{t^{-1}}{-1} + C = -3t^{-1} + C \).
This step transforms the integrand by systematically following the exponent rules, allowing us to find the antiderivative.
Constant of Integration
Whenever we integrate a function, we include a 鈥�+ C鈥� in our result. This is known as the Constant of Integration. It's an essential part of the process because it accounts for any constant we could have had in the original function before differentiation.
Why is this important? Imagine that \(F(t)\) and \(F(t) + D\) (where \(D\) is any constant) could both be antiderivatives of the same differential function. Differentiating a constant gives zero, so any constant added to the antiderivative does not change the differentiation result.
  • The presence of \(C\) in the antiderivative \(-\frac{3}{t} + C\) signifies the infinite possibilities of the original function.
  • The plus \(C\) ensures we include all potential vertical shifts in the family of curves described by the antiderivative.
Thus, always remember that the Constant of Integration captures the inherent "flexibility" in solutions for antiderivatives.
Differentiation Verification
Once we have found an antiderivative, the final step is to check our work through Differentiation Verification. This involves differentiating the antiderivative to see if we return to the original function. For our found antiderivative \(-\frac{3}{t} + C\), let's differentiate:
  • The derivative of \(-\frac{3}{t}\) is calculated using the power rule for derivatives. Rewrite it first as \(-3t^{-1}\).
  • Differentiate: \(-3 \cdot (-1) \cdot t^{-2} = \frac{3}{t^2}\).
  • Note that \(C\), being a constant, vanishes since its derivative is zero.
By differentiating and obtaining the original function \(\frac{3}{t^2}\), we verify the correctness of our antiderivative. This step is crucial as it provides confidence that the antiderivative calculation was performed correctly.

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