Problem 3 - Thomas Calculus

Question

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }-3 x^{-4} \quad \text { b. } x^{-4} \quad \text { c. } x^{-4}+2 x+3$$

Step-by-Step Solution

Answer

Antiderivatives are: a. \(x^{-3} + C\), b. \(-\frac{1}{3}x^{-3} + C\), c. \(-\frac{1}{3}x^{-3} + x^2 + 3x + C\).

Step 1: Recognize the Function for Part a

First, look at the function given: \(-3x^{-4}\). We need to find an antiderivative for this function.

Step 2: Apply the Power Rule for Integration to Part a

The power rule for integration states that to integrate a term of the form \(ax^n\), the antiderivative is given by \(\frac{a}{n+1}x^{n+1}\) given that \(n eq -1\). For the term \(-3x^{-4}\), apply the rule:\[\int -3x^{-4} \, dx = -\frac{3}{-3}x^{-3} = x^{-3} + C\] Here, \(C\) represents the constant of integration.

Step 3: Check Answer for Part a by Differentiation

Differentiate \(x^{-3}+C\).The derivative of \(x^{-3}\) is \(-3x^{-4}\), which matches the original function, confirming the antiderivative is correct.

Step 4: Recognize the Function for Part b

For part b, the function is given as \(x^{-4}\). Identify that we need to integrate this function.

Step 5: Apply the Power Rule to Part b

Apply the power rule again to \(x^{-4}\):\[\int x^{-4} \, dx = \frac{1}{-3}x^{-3} = -\frac{1}{3}x^{-3} + C\]

Step 6: Check Answer for Part b by Differentiation

Differentiate \(-\frac{1}{3}x^{-3} + C\).The derivative of \(-\frac{1}{3}x^{-3}\) is \(x^{-4}\), matching the original function.

Step 7: Recognize the Function for Part c

Part c presents the function \(x^{-4}+2x+3\). We will integrate each term separately.

Step 8: Integrate Each Term of Part c Separately

Using the power rule, integrate each term of the function \(x^{-4} + 2x + 3\):\[\int x^{-4} \, dx = -\frac{1}{3}x^{-3}\]\[\int 2x \, dx = 2\cdot\frac{1}{2}x^2 = x^2\]\[\int 3 \, dx = 3x\]

Step 9: Combine the Antiderivatives for Part c

Now, add the results together, including the constant of integration:\[-\frac{1}{3}x^{-3} + x^2 + 3x + C\]

Step 10: Check Answer for Part c by Differentiation

Differentiate the combined antiderivative\(-\frac{1}{3}x^{-3} + x^2 + 3x + C\):This results in \(x^{-4} + 2x + 3\), confirming it matches the original function.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a handy tool when you need an antiderivative of a term like \( ax^n \). The rule simplifies the integration process by providing a straightforward formula: \( \int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C \), where \( n eq -1 \). When you apply this formula, you increase the exponent \( n \) by 1 and divide the coefficient \( a \) by the new exponent.Let's see how this works with examples:

• For \(-3x^{-4}\), applying the rule gives us the antiderivative \( x^{-3} + C \).
• Similarly, for \(x^{-4}\), the antiderivative becomes \( -\frac{1}{3}x^{-3} + C \).
• For combined terms like \(x^{-4}+2x+3\), apply the power rule to each term individually.

This rule is essential for finding antiderivatives, which are the inverse operation of derivatives.

Constant of Integration

When integrating a function, you'll notice we often add a \( + C \) at the end. This \( C \) is called the 'Constant of Integration.' Why is it important?

• Antiderivatives aren't unique. Adding \( C \) accounts for all possible antiderivatives of a function.
• Functions with different constants of integration have the same derivative but are vertically shifted on a graph.
• This constant ensures the general solution; without it, we could miss valid solutions.

For instance, in the antiderivative \( x^{-3} + C \), \( C \) can be any constant, representing an infinite set of solutions to the differential equation.

Differentiation to Check Antiderivative

Once you've found an antiderivative, it's essential to verify it's correct. Differentiation is the process used to check your work. By differentiating your antiderivative, you should arrive back at the original function if your antiderivative is correct.Steps to Check:

• Differentiate the function that you found as the antiderivative.
• If the derivative matches the starting function, the antiderivative is verified.

For example:

• Given the antiderivative \( x^{-3} + C \), differentiate \( x^{-3} \). You'll get \(-3x^{-4}\), which matches the original function \(-3x^{-4}\).
• This process confirms that the integration and the addition of the constant of integration were done correctly.