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. a. \(-3 x^{-4} \quad\) b. \(x^{-4} \quad\) c. \(x^{-4}+2 x+3\)

Step-by-Step Solution

Answer

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

Step 1: Identify the function form

Observe the given function: - a. \(-3x^{-4}\)- b. \(x^{-4}\)- c. \(x^{-4} + 2x + 3\)Each function can be represented as a polynomial with terms in the form \(ax^n\). To find the antiderivative, we will use the basic antiderivative rules for polynomials.

Step 2: Apply integration rules to part (a)

For part (a), the function is \(-3x^{-4}\).The antiderivative of a function \(x^n\) is \(\frac{x^{n+1}}{n+1}\), plus \(C\), a constant of integration.Thus, the antiderivative is:\[\int -3x^{-4} \, dx = -3 \cdot \frac{x^{-3}}{-3} + C = x^{-3} + C\]

Step 3: Perform differentiation check for part (a)

Verify by differentiation:Differentiating \(x^{-3} + C\), we get:\[\frac{d}{dx}(x^{-3} + C) = -3x^{-4}\]This matches the original function \(-3x^{-4}\), confirming our antiderivative is correct.

Step 4: Apply integration rules to part (b)

For part (b), the function is \(x^{-4}\).The antiderivative is found in a similar way:\[\int x^{-4} \, dx = \frac{x^{-3}}{-3} + C = -\frac{1}{3}x^{-3} + C\]

Step 5: Perform differentiation check for part (b)

Verify by differentiation:Differentiating \(-\frac{1}{3}x^{-3} + C\), we get:\[\frac{d}{dx}(-\frac{1}{3}x^{-3} + C) = x^{-4}\]This matches the original function \(x^{-4}\), confirming our antiderivative is correct.

Step 6: Apply integration rules to part (c)

For part (c), the function is \(x^{-4} + 2x + 3\).Integrate each term separately:- For \(x^{-4}\), follow the same steps from part (b): \(-\frac{1}{3}x^{-3}\).- For \(2x\), the antiderivative is \(x^2\).- For constant 3, the antiderivative is \(3x\).Combine results:\[\int (x^{-4} + 2x + 3) \, dx = -\frac{1}{3}x^{-3} + x^2 + 3x + C\]

Step 7: Perform differentiation check for part (c)

Verify by differentiation:Differentiating \(-\frac{1}{3}x^{-3} + x^2 + 3x + C\), we get:\[\frac{d}{dx}(-\frac{1}{3}x^{-3} + x^2 + 3x + C) = x^{-4} + 2x + 3\]This confirms our antiderivative is correct.

Key Concepts

Integration Rules

Integration is like the reverse process of differentiation. Instead of finding the rate of change as differentiation does, integration helps us find an antiderivative. An antiderivative is a function whose derivative gives us back the original function. To find it, knowing the integration rules is key.

• The basic rule for integrating a power function \(x^n\) is to add one to the exponent \(n\), then divide by the new exponent. This gives us the antiderivative: \(\frac{x^{n+1}}{n+1} + C\).
• Each function term can be integrated separately, making it easier to deal with polynomial functions.
• A constant \(C\) is always added as an antiderivative because integration is not a single-valued process. There are infinite antiderivatives depending on this constant.

This is reflected in the exercise, where each part uses the integration rule to find the antiderivative.

Polynomial Functions

Polynomial functions are a sum of terms, where each term is a variable raised to a non-negative integer exponent, like \(ax^n\). Antiderivatives of polynomial functions follow straightforward integration rules:

• Each term of the polynomial can be integrated individually. This makes finding antiderivatives of polynomials simpler.
• To integrate, increase the exponent by one and divide by this new exponent.
• Combining all integrated terms then gives us the polynomial's antiderivative.

For example, the function in part (c) combines multiple polynomial terms—an inverse power, a linear term, and a constant term. Each is treated independently, and then they are summed together.

Differentiation Verification

Verification by differentiation confirms if an antiderivative is correct. This serves as a helpful check when solving integration problems.

• Differentiating the antiderivative should lead back to the original function.
• Use basic differentiation rules for checking, like \(\frac{d}{dx}(x^n) = nx^{n-1}\).
• If the derivative matches the original function, then the antiderivative is verified.

In the original exercise, differentiation checks verify that each antiderivative found is correct, such as differentiating \(x^{-3} + C\) to confirm it equals \(-3x^{-4}\), matching the original function and proving the solution is accurate.