Problem 27 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int(\sqrt{x}+\sqrt[3]{x}) d x $$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \).

Step 1: Identify the Integral Components

We are given the integral \( \int (\sqrt{x} + \sqrt[3]{x}) \, dx \). This consists of two terms: \( \sqrt{x} \) and \( \sqrt[3]{x} \). We can write these in power form as \( x^{1/2} \) and \( x^{1/3} \), respectively.

Step 2: Integrate Each Term Separately

Using the power rule for integration, which is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), we integrate each term separately:- For \( \int x^{1/2} \, dx \), the result is \( \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2} \).- For \( \int x^{1/3} \, dx \), the result is \( \frac{x^{1/3 + 1}}{1/3 + 1} = \frac{x^{4/3}}{4/3} = \frac{3}{4}x^{4/3} \).

Step 3: Combine the Results and Add Constant of Integration

Combine the integrated results of both terms:\[ \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \]Here, \( C \) is the constant of integration, representing the family of antiderivatives.

Step 4: Verify the Solution by Differentiation

To verify, differentiate \( \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \):- The derivative of \( \frac{2}{3}x^{3/2} \) is \( \frac{2}{3} \cdot \frac{3}{2}x^{1/2} = x^{1/2} \).- The derivative of \( \frac{3}{4}x^{4/3} \) is \( \frac{3}{4} \cdot \frac{4}{3}x^{1/3} = x^{1/3} \).Adding them gives \( x^{1/2} + x^{1/3} \), which matches the integrand, confirming the solution is correct.

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is a fundamental concept in calculus. It represents the reverse process of differentiation and is used to find a function whose derivative is the given function. When we integrate, we are essentially finding a function that describes the area under the curve of the given function. This process does not provide a specific solution but rather a general form that includes a constant of integration.

In mathematical notation, the indefinite integral of a function \( f(x) \) is written as \( \int f(x) \, dx \). This indicates that we are integrating the function \( f(x) \) with respect to \( x \). In our exercise, the integral \( \int (\sqrt{x} + \sqrt[3]{x}) \, dx \) is an example where we seek the antiderivative of the given function. This operation helps us understand how the derivative of a function works in reverse, allowing flexibility in manipulating equations.

Power Rule for Integration

The power rule for integration is a handy tool when dealing with polynomial expressions or terms raised to a power. It is expressed as \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This formula tells us how to handle integration when the function is a simple power of \( x \).

In the given problem, we are integrating \( \sqrt{x} \) and \( \sqrt[3]{x} \), which translate to \( x^{1/2} \) and \( x^{1/3} \) in power form. Applying the power rule separately:

• For \( x^{1/2} \), the integral becomes \( \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{2}{3}x^{3/2} \).
• For \( x^{1/3} \), the integral becomes \( \frac{x^{1/3 + 1}}{1/3 + 1} = \frac{3}{4}x^{4/3} \).

By applying this rule, we handle fractions in the powers seamlessly, converting them into more manageable terms that resemble polynomials.

Integration Verification

After finding the indefinite integral, it is crucial to verify the solution by differentiation. This step ensures that the integration process has been executed correctly. If we correctly integrated the function, differentiating it should return us to the original function.

For our example, we found the antiderivative \( \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \). To verify:

• Differentiate \( \frac{2}{3}x^{3/2} \). The result is \( \frac{2}{3} \times \frac{3}{2}x^{1/2} = x^{1/2} \).
• Differentiate \( \frac{3}{4}x^{4/3} \). This results in \( \frac{3}{4} \times \frac{4}{3}x^{1/3} = x^{1/3} \).

Summing these derivatives gives us the original function, \( x^{1/2} + x^{1/3} \). This confirms the accuracy of our integration.

Constant of Integration

When dealing with indefinite integrals, the constant of integration \( C \) plays a pivotal role. This constant arises because the process of differentiation does not give us the information about any constants that were present in the original function.

Without \( C \), we would only have one particular solution from the infinite set of possible solutions. For example, if \( F(x) \) is any antiderivative of \( f(x) \), then \( F(x) + C \) is also an antiderivative for any constant value of \( C \). Each value of \( C \) gives a different function, illustrating the infinite nature of solutions to an indefinite integral.

• The presence of \( C \) reflects all possible vertical shifts of the antiderivative.
• When differentiating, the constant disappears, which is why starting integration from a known value or initial condition can determine \( C \).

Hence, always remember to include \( C \) when solving indefinite integrals, encapsulating the entirety of potential solutions.