Problem 63 - 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. $$\int 3 x^{\sqrt{3}} d x$$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{3}{\sqrt{3}+1} \cdot x^{\sqrt{3}+1} + C \).

Step 1: Understanding the Integral

We need to find the most general antiderivative of the function. We have the integral \( \int 3x^{\sqrt{3}} \, dx \). The integrand is a power function, so we will apply the power rule for integration.

Step 2: Applying the Power Rule for Integration

The power rule for integration states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). For our integral, \( n = \sqrt{3} \). Thus, the antiderivative is \( \frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C \).

Step 3: Solution to the Integral

By applying the power rule, the indefinite integral of \( \int 3x^{ \sqrt{3}} \, dx \) is \( \frac{3}{\sqrt{3}+1} \cdot x^{\sqrt{3}+1} + C \), where \( C \) is the constant of integration.

Step 4: Verify by Differentiation

Now we differentiate our solution to check correctness. Differentiate \( \frac{3}{\sqrt{3}+1} \cdot x^{\sqrt{3}+1} + C \) with respect to \( x \). Applying the power rule for differentiation, \( \left( \sqrt{3}+1 \right) \cdot \frac{3x^{\sqrt{3}}}{\sqrt{3}+1} = 3x^{\sqrt{3}} \), which matches our original integrand.

Key Concepts

Antiderivative

An antiderivative is a function that reverses the process of differentiation. It essentially "undoes" differentiation to retrieve the original function. In other words, if you differentiate an antiderivative, you should arrive at the original function.

For any function, its antiderivative is not unique. There can be many functions, all differing by a constant value, which qualify as antiderivatives. This leads to the concept of an indefinite integral. Indefinite integrals do not have limits of integration and instead focus on finding the general formula for all possible antiderivatives of a function.

• Indefinite Integral is represented by the symbol \( \int \).
• An arbitrary constant \( C \) is added, representing all possible vertical shifts of the antiderivative.

Applying these principles to our given exercise, the antiderivative involves identifying the form of the power function and finding the corresponding indefinite integral using appropriate rules.

Power Rule for Integration

The power rule for integration is a straightforward and essential tool for finding antiderivatives, particularly for power functions. It is primarily used when integrating expressions of the form \( x^n \).

The rule states:

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

This rule is directly applied in our example, where the function is \( 3x^{\sqrt{3}} \). By substituting \( n = \sqrt{3} \) into the power rule formula, we calculate:

• \( \int 3x^{\sqrt{3}} \, dx = \frac{3x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C \)

This provides the most general antiderivative for our function, showcasing how the power rule allows us to efficiently integrate power expressions by systematically increasing the exponent and dividing by the new value.

Differentiation Verification

One of the best ways to ensure that an indefinite integral has been computed correctly is to differentiate the resulting expression. Differentiation verification helps check the correctness by taking the derivative and seeing if it matches the original integrand.

In our example, we found the antiderivative to be \( \frac{3}{\sqrt{3}+1} \cdot x^{\sqrt{3}+1} + C \). To verify, follow these steps:

• Differentiate \( \frac{3}{\sqrt{3}+1} \cdot x^{\sqrt{3}+1} \).
• The derivative, using the power rule for differentiation, should yield: \( (\sqrt{3}+1) \cdot \frac{3x^{\sqrt{3}}}{\sqrt{3}+1} \).
• This simplifies to: \( 3x^{\sqrt{3}} \), which is the original integrand.

By achieving this result, the process verifies that the solution is correct, providing confidence that the integration was carried out properly.