Problem 22 - University Calculus: Early Transcendentals

Question

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(x^{\sqrt{3}}\) b. \(x^{\pi}\) c. \(x^{\sqrt{2}-1}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding Antiderivatives

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \). To find an antiderivative of expressions like \( x^n \), you apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is a constant.

Step 2: Solve for \( x^{\sqrt{3}} \)

For the function \( x^{\sqrt{3}} \), apply the power rule: \[ \int x^{\sqrt{3}} \, dx = \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C \]. Check by differentiating: \( \frac{d}{dx} \left( \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} \right) = x^{\sqrt{3}} \), confirming the antiderivative.

Step 3: Solve for \( x^{\pi} \)

Using the same method, for \( x^{\pi} \), the antiderivative is \[ \int x^{\pi} \, dx = \frac{x^{\pi + 1}}{\pi + 1} + C \]. Verify by differentiation: \( \frac{d}{dx} \left( \frac{x^{\pi + 1}}{\pi + 1} \right) = x^{\pi} \), confirming it is correct.

Step 4: Solve for \( x^{\sqrt{2}-1} \)

Now for \( x^{\sqrt{2}-1} \), apply the rule: \[ \int x^{\sqrt{2} - 1} \, dx = \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \], as \( n + 1 = \sqrt{2} \). Differentiate to check: \( \frac{d}{dx} \left( \frac{x^{\sqrt{2}}}{\sqrt{2}} \right) = x^{\sqrt{2}-1} \). It confirms the antiderivative is correct.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus. It helps us find antiderivatives of functions that have the form \( x^n \), where \( n \) is any real number. This rule states that if we have a function \( x^n \), then its antiderivative is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( C \) is the constant of integration. It's important to note that this rule applies when \( n eq -1 \). This is because the formula would involve division by zero, which is undefined. By following this rule, we can determine the area under the curve for functions that are expressed as \( x^n \), making it a go-to method when computing integrals involving exponents.
Applying this rule involves a simple, consistent step: you increase the exponent of \( x \) by one, and divide by this new exponent.

• Example 1: For \( x^{\sqrt{3}} \), add 1 to \( \sqrt{3} \) to get \( \sqrt{3} + 1 \), then divide by this result, giving you an antiderivative \( \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C \).
• Example 2: With \( x^{\pi} \), the exponent increases to \( \pi + 1 \), yielding \( \frac{x^{\pi + 1}}{\pi + 1} + C \).

Differentiation

Differentiation is the process of finding the derivative of a function, and it tells us how the function changes at any given point. In the context of verifying antiderivatives, differentiation plays a critical role. To confirm that a function \( F(x) \) is indeed the antiderivative of \( f(x) \), you differentiate \( F(x) \). If \( F'(x) = f(x) \), then \( F(x) \) is a correct antiderivative.
For the exercise at hand, after finding the antiderivatives using the power rule, we used differentiation to check our work. For instance, by taking the derivative of \( \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} \), we found it equals \( x^{\sqrt{3}} \). This confirms our integration was accurate.

• This process involves applying the power rule for differentiation, where \( \frac{d}{dx} (x^n) = nx^{n-1} \).
• It's a looping check, where the derivative gets us back to the original function, ensuring the integration process was done correctly.

Functions with Exponent

Functions with exponents represent a broad category of mathematical expressions and are frequently encountered in calculus. They are characterized by variable bases raised to some power, such as \( x^a \). These functions are essential in various fields, including physics, engineering, and economic growth models.
When dealing with such functions, understanding the behavior of the exponent is crucial. It dictates the growth rate and overall shape of the graph.

• If the exponent is positive, the function grows as \( x \) increases.
• If it is negative, the function decreases as \( x \) increases, reflecting the mirror behavior of positive exponents.

In integration, as seen with functions like \( x^{\pi} \) or \( x^{\sqrt{2}-1} \), handling these exponents is integral to finding the antiderivative. The power rule directly applies, making it easy to deal with fractional and irrational exponents by treating them just like integers during integration. This seamless treatment explains their prevalence in various mathematical solutions.