Problem 35 - 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. $$\int(\sqrt{x}+\sqrt[3]{x}) d x$$

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: Rewrite Radicals as Exponents

The expression contains square roots and cube roots. Rewrite these in terms of exponents: \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). So the integral becomes \[ \int(x^{1/2} + x^{1/3}) \, dx. \]

Step 2: Apply the Power Rule for Integration

The power rule for integration tells us that \( \int x^n \ dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Applying this to each term:- For \( x^{1/2} \), integrable as: \[ \int x^{1/2} \, dx = \frac{x^{1/2+1}}{1/2+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}. \]- For \( x^{1/3} \), integrable as: \[ \int x^{1/3} \, dx = \frac{x^{1/3+1}}{1/3+1} = \frac{x^{4/3}}{4/3} = \frac{3}{4}x^{4/3}. \]

Step 3: Combine the Integrated Terms

Now that each term is integrated, combine them back together in one expression, adding the constant of integration \( C \). The integral becomes:\[ \int(x^{1/2} + x^{1/3}) \, dx = \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C. \]

Step 4: Verify by Differentiating

Verify the result by differentiating the antiderivative. Differentiate \( \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \):- Derivative of \( \frac{2}{3}x^{3/2} \) is \( \frac{2}{3} \cdot \frac{3}{2}x^{3/2-1} = x^{1/2} \).- Derivative of \( \frac{3}{4}x^{4/3} \) is \( \frac{3}{4} \cdot \frac{4}{3}x^{4/3-1} = x^{1/3} \).Adding these derivatives: \( x^{1/2} + x^{1/3} \), which matches the original integrand.

Key Concepts

Power Rule for Integration

Integration is the operation inverse to differentiation. The power rule for integration is one of the simplest and most useful rules when dealing with polynomials and terms with variable bases raised to a power. The rule states that for a function of the form \( x^n \), the integral is given by:

• \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Here, \( C \) is the constant of integration, accounting for any constant term lost during differentiation.

To apply the power rule, it's crucial that \( n eq -1 \) because dividing by zero is undefined. In our steps, the power rule allowed us to integrate terms like \( x^{1/2} \) and \( x^{1/3} \) easily by adding one to the exponent, then dividing by the new exponent.

Using the power rule is straightforward once you've rewritten terms to clear radicals or fractions. Rewriting is often a necessary precursor to applying this rule effectively.

Radical Expressions

Radical expressions often deal with roots, like square roots or cube roots. These are expressions that involve the radical symbol (√). Handling these can be tricky, but converting them into expressions with exponents simplifies integration.

A root such as \( \sqrt{x} \) can be rewritten with a fractional exponent: \( \sqrt{x} = x^{1/2} \). Similarly, a cube root can be expressed as \( \sqrt[3]{x} = x^{1/3} \). This conversion is extremely useful in calculus since it allows us to use the power rule for integration and differentiation easily.

After converting radicals to exponents, the rest of the integration becomes a matter of applying the power rule, which is much easier to handle. Remember that after completing these types of problems, always check back to ensure that the rewritten terms still represent the original problem.

Antiderivative Verification

After finding an antiderivative, it's essential to verify it is correct by differentiating back to the original function. This helps confirm that no mistakes were made during integration.

To verify an antiderivative, differentiate it using the power rule for differentiation.

• The power rule for differentiation is: \( \frac{d}{dx} x^n = n \cdot x^{n-1} \).
• For example, for \( \frac{2}{3}x^{3/2} \), differentiating gives us \( \frac{2}{3} \cdot \frac{3}{2}x^{1/2} = x^{1/2}, \)
• For \( \frac{3}{4}x^{4/3} \), differentiating gives \( \frac{3}{4} \cdot \frac{4}{3}x^{1/3} = x^{1/3} \).

After verifying, the differentiated terms should match the original integrand, confirming the correctness of your antiderivative. Using this method helps ensure that the integration process was accurate and can build confidence in similar future problems.