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Chapter 4: Problem 1

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(2 x\) b. \(x^{2}\) c. \(x^{2}-2 x+1\)

Short Answer

Expert verified
The antiderivatives are \( x^2 + C \), \( \frac{1}{3}x^3 + C \), and \( \frac{1}{3}x^3 - x^2 + x + C \).

Step by step solution

01

Understanding the Problem

We need to find the antiderivatives of the given functions. An antiderivative of a function is another function whose derivative is equal to the original function. The general antiderivative includes a constant of integration, denoted as +C.
02

Antiderivative of 2x

To find the antiderivative of the function \( 2x \), we use the power rule for integration: if \( f(x) = ax^n \), then its antiderivative is \( \frac{a}{n+1}x^{n+1} + C \). For \( 2x \), it is equivalent to \( 2x^1 \), so the antiderivative is \( x^2 + C \).
03

Antiderivative of x^2

The function is \( x^2 \). Using the power rule for integration, we increase the power by one and divide by the new power: \( \frac{1}{2+1}x^{2+1} + C = \frac{1}{3}x^{3} + C \).
04

Antiderivative of x^2 - 2x + 1

We find the antiderivative of each term separately: \( \int x^2 \, dx = \frac{1}{3}x^3 + C_1 \), \( \int -2x \, dx = -x^2 + C_2 \), and \( \int 1 \, dx = x + C_3 \). Combining these gives \( \frac{1}{3}x^3 - x^2 + x + C \) where \( C \) is the constant of integration.
05

Checking by Differentiation

We verify by differentiating our antiderivatives. The derivative of \( x^2 + C \) is \( 2x \), the derivative of \( \frac{1}{3}x^3 + C \) is \( x^2 \), and the derivative of \( \frac{1}{3}x^3 - x^2 + x + C \) is \( x^2 - 2x + 1 \). Each matches the original function, confirming our antiderivatives are correct.

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Key Concepts

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

Power Rule for Integration
The power rule for integration is a useful tool when computing antiderivatives. It's the counterpart of the power rule in differentiation. Whenever you see a function in the form of \( ax^n \), you can find its antiderivative quite easily.
  • Start by increasing the exponent by one. Thus, \( n \) becomes \( n+1 \).
  • Then, divide the entire term by this new exponent. This transforms the function to \( \frac{a}{n+1}x^{n+1} \).
This rule allows us to handle polynomials efficiently. As seen in the solution:For \( 2x \) (or \( 2x^1 \)), applying the power rule gives \( x^2 + C \). For \( x^2 \), we integrate to \( \frac{1}{3}x^3 + C \). Each term in a polynomial can be handled separately, making the process straightforward and logical. Remember this rule whenever you're working with similar functions, as it simplifies integration greatly.
Constant of Integration
When finding an antiderivative, it's crucial to remember the constant of integration, noted as \( +C \). This constant accounts for the "family" of possible antiderivatives.
  • Differentiating a constant results in zero. Therefore, any constant added during integration does not affect the validation process through differentiation.
  • The constant is important because an indefinite integral fundamentally represents all antiderivatives of a function.
During exercises, such as in finding the antiderivative of \( 2x \), \( x^2 \), or \( x^2 - 2x + 1 \), this constant ensures we recognize all potential functions whose derivative aligns with the original. Omitting \( +C \) can lead to incomplete solutions and sometimes incorrect mathematical interpretations.
Differentiation Check
After finding an antiderivative, performing a differentiation check is an effective way to verify your solution. This involves differentiating your antiderivative to see if you retrieve the original function.
  • If successful, the antiderivative is most likely correct. For instance, differentiating \( x^2 + C \) results in \( 2x \), confirming the antiderivative matches the original function.
  • Checking with differentiation exposes calculation errors or missed constants. It鈥檚 a practical safeguard to ensure that all antiderivatives are correct.
For students, consistently retrying problems by using differentiation as a check can build confidence and understanding in both integration and differentiation. It鈥檚 a simple trick that adds robustness to your mastery of calculus.

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Most popular questions from this chapter

Verify the formulas by differentiation. $$\int(7 x-2)^{3} d x=\frac{(7 x-2)^{4}}{28}+C$$

Find the absolute maximum and the absolute minimum values of \(f(x)=e^{x}-2 x\) on [0,1].

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{\sec ^{3} \theta}{3}+C\) b. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \tan ^{2} \theta+C\) c. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \sec ^{2} \theta+C\)

Verify the formulas by differentiation. $$\int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C$$

Verify the formulas by differentiation. $$\int \sec ^{2}(5 x-1) d x=\frac{1}{5} \tan (5 x-1)+C$$
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