Problem 27 - Thomas Calculus in SI Units

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: Simplify the Expression

Rewrite the integrand by expressing the square root and cube root in terms of exponents: \[ \sqrt{x} = x^{1/2} \quad \text{and} \quad \sqrt[3]{x} = x^{1/3}. \]Thus, the integral becomes:\[ \int (x^{1/2} + x^{1/3}) \, dx. \]

Step 2: Use the Power Rule for Integration

Apply the power rule for integration, \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \]where \(C\) is the constant of integration.Integrating each term separately:\[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C_1 = \frac{x^{3/2}}{3/2} + C_1 = \frac{2}{3} x^{3/2} + C_1, \]\[ \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C_2 = \frac{x^{4/3}}{4/3} + C_2 = \frac{3}{4} x^{4/3} + C_2, \]Combining these results, the antiderivative is:\[ \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C. \]

Step 3: Check by Differentiation

Differentiate the antiderivative to ensure it matches the original integrand. The derivative of the result is:\( \frac{d}{dx} \left( \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C \right) \).Apply the power rule for differentiation:\[ \frac{d}{dx} \left( \frac{2}{3}x^{3/2} \right) = \frac{2}{3} \cdot \frac{3}{2} x^{1/2} = x^{1/2}, \]\[ \frac{d}{dx} \left( \frac{3}{4}x^{4/3} \right) = \frac{3}{4} \cdot \frac{4}{3} x^{1/3} = x^{1/3}. \]Adding these results gives:\[ x^{1/2} + x^{1/3}, \]which matches the original integrand.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus, making it easy to integrate polynomial functions. This rule is applied to find the antiderivative or indefinite integral of a function of the form \(x^n\), where \(n\) is any real number except \(-1\). In simple terms, the rule states:

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

Here, \(C\) represents the constant of integration, which is crucial because indefinite integrals can have many solutions that differ by a constant.

When applying this rule, just add 1 to the exponent \(n\) and then divide by the new exponent. For example, to integrate \(x^{1/2}\), you add 1 to the power to get \(3/2\) and then divide by \(3/2\), yielding \(\frac{2}{3}x^{3/2} + C\). Similarly, for \(x^{1/3}\), adding 1 gives \(4/3\) and dividing by \(4/3\) gives \(\frac{3}{4}x^{4/3} + C\).
Knowing this rule, it becomes simple to compute integrals involving roots and fractional exponents by expressing them as power functions.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function you started with. Finding antiderivatives is an essential part of solving integrals when only basic rules are applied, such as the power rule.

In practical terms, if you are given the function \(f(x) = \sqrt{x} + \sqrt[3]{x}\), finding its antiderivative means looking for a function \(F(x)\) such that when differentiated, it yields \(f(x)\).

To identify the antiderivative:

• Recognize each term in the function to use the appropriate integration rule, like the power rule.
• Apply the rule separately to life each component, combining them appropriately.
• Don't forget to add \(C\), representing all the possible constant differences between antiderivatives.
  For instance, in our exercise, the antiderivative of \(\sqrt{x}\) is \(\frac{2}{3}x^{3/2}\) and for \(\sqrt[3]{x}\), it is \(\frac{3}{4}x^{4/3}\), combined as \(\frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C\).

Differentiation Check

Once you've found a potential antiderivative, performing a differentiation check is a key verification step. This process ensures that the antiderivative you obtained actually represents the original function you integrated. Here's how you check your work:

Begin by differentiating the antiderivative you found. Remember, the derivative operation essentially reverses the integration process. For example, for the function \(F(x) = \frac{2}{3}x^{3/2} + \frac{3}{4}x^{4/3} + C\), compute its derivative:

- Differentiate each term separately:
- For \(\frac{2}{3}x^{3/2}\), apply the power rule for differentiation:
\(\frac{d}{dx} \left( \frac{2}{3}x^{3/2} \right) = x^{1/2}\).
- For \(\frac{3}{4}x^{4/3}\), do the same:
\(\frac{d}{dx} \left( \frac{3}{4}x^{4/3} \right) = x^{1/3}\).
- The constant \(C\) disappears because the derivative of a constant is zero.

Combine these results; if the result matches the original function \(\sqrt{x} + \sqrt[3]{x}\), then your solution is confirmed correct.
This step may seem like extra work, but it's an essential practice for ensuring your integrals are solved accurately.