Problem 64 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int x^{\sqrt{2}-1} d x$$

Step-by-Step Solution

Answer

\( \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \) is the most general antiderivative.

Step 1: Identify the Function’s Form

The given integral is \( \int x^{\sqrt{2}-1} \, dx \). This is a power function of the form \( x^n \), where \( n = \sqrt{2} - 1 \). We can use the power rule for integration for this function.

Step 2: Apply the Power Rule for Integration

The power rule for integration is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Applying it to our integral, we get:\[ \int x^{\sqrt{2}-1} \, dx = \frac{x^{(\sqrt{2}-1)+1}}{\sqrt{2}-1+1} + C \]

Step 3: Simplify the Expression

Now simplify the expression:First, simplify the exponent:\( (\sqrt{2} - 1) + 1 = \sqrt{2} \), so the power of \( x \) becomes \( \sqrt{2} \).So, the integral becomes:\[ \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \]

Step 4: Check by Differentiating the Antiderivative

Differentiate \( \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \):\[ \frac{d}{dx} \left( \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \right) = \frac{1}{\sqrt{2}} \cdot \sqrt{2} \cdot x^{\sqrt{2}-1} = x^{\sqrt{2}-1} \]Since differentiating the antiderivative gives back the original function, the solution is verified.

Key Concepts

Integration

Integration is one of the fundamental operations in calculus, often viewed as the reverse process of differentiation. It helps in finding the antiderivative of a function. An antiderivative of a function is another function whose derivative is the original function. For example, if you have a function that describes velocity, integrating it helps you find the function that describes position.
There are different methods to perform integration, such as substitution, integration by parts, and using integration rules like the power rule. In the problem given, we use the concept of indefinite integrals. These do not have specified limits; therefore, the result includes a constant of integration, denoted by a "+ C".
Indefinite integrals typically take the form:

• \( \int f(x) \, dx = F(x) + C \)

where \( F(x) \) is the antiderivative and \( C \) is the constant of integration.

Power Rule

The power rule for integration is a straightforward formula used to integrate functions of the form \( x^n \). It states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n \) cannot be equal to -1 because that would involve division by zero.
Using the power rule simplifies the integration of polynomials and functions with constant exponents. In our problem, we use the power rule to solve \( \int x^{\sqrt{2}-1} \, dx \). By considering the exponent as \( n = \sqrt{2} - 1 \), we can directly apply the power rule which results in the expression:
\[ \int x^{\sqrt{2}-1} \, dx = \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \] The power rule helps to quickly determine this expression by employing a simple adjustment to the exponent, adding one and dividing by the new exponent.

Differentiation Check

Once you find an antiderivative through integration, it's crucial to verify it by differentiating. This ensures you return to the original function, confirming the correctness of your integration. This step is called a differentiation check.
To perform a differentiation check, differentiate the antiderivative using standard differentiation rules. If you end up with your original function, you have successfully found the antiderivative.
For example, differentiating \( \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \) involves applying the power rule of differentiation:

• The derivative of \( x^{\sqrt{2}} \) is \( \sqrt{2} \cdot x^{\sqrt{2} - 1} \)

Thus, \[\frac{d}{dx} \left( \frac{x^{\sqrt{2}}}{\sqrt{2}} + C \right) = \frac{1}{\sqrt{2}} \cdot \sqrt{2} \cdot x^{\sqrt{2}-1} = x^{\sqrt{2}-1}\] Seeing that this equals the original integrand, \( x^{\sqrt{2}-1} \), reassures us that our integration was done correctly. The differentiation check is a valuable step that enhances confidence in the integration process.