Problem 81 - Applied Calculus

Question

Find an antiderivative and use differentiation to check your answer. $$g(x)=x \sqrt{x}$$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{2}{5} x^{5/2} + C \), and it checks out upon differentiation.

Step 1: Rewrite the Function

The given function is \( g(x) = x \sqrt{x} \). To find an antiderivative, first express \( \sqrt{x} \) as a power of \( x \). This gives us \( g(x) = x \cdot x^{1/2} = x^{3/2} \).

Step 2: Apply the Power Rule for Integration

Use the power rule to find the antiderivative of \( x^{3/2} \). The power rule states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Here, \( n = 3/2 \), so we get:\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} + C = \frac{x^{5/2}}{5/2} + C = \frac{2}{5} x^{5/2} + C \]

Step 3: Differentiate to Check the Solution

Differentiate the antiderivative \( G(x) = \frac{2}{5} x^{5/2} + C \) to verify it equals the original function \( g(x) \). Differentiating, we have:\[ G'(x) = \frac{d}{dx} \left( \frac{2}{5} x^{5/2} + C \right) = \frac{2}{5} \cdot \frac{5}{2} x^{5/2 - 1} = x^{3/2} \]Since \( G'(x) = g(x) = x^{3/2} \), our solution is verified.

Key Concepts

Antiderivative

The antiderivative, also known informally as the indefinite integral, is essentially the reverse process of differentiation. In simple terms, if you have a function and you find its derivative, an antiderivative helps you go back to the original function. Finding an antiderivative requires integrating the given function.

For instance, if we are given a function like \( g(x) = x \sqrt{x} \), the first step is to simplify it using exponent rules. We express \( \sqrt{x} \) in terms of a power, so it becomes \( g(x) = x^{3/2} \). By integrating this, we use the power rule and find its antiderivative. Thus, the antiderivative of \( g(x) \) is a function such that when we differentiate it, we get \( g(x) \) back.

Power Rule

The power rule for integration is a handy formula that simplifies finding antiderivatives for polynomial expressions. It states that if you need to integrate \( x^n \), where \( n eq -1 \), the formula is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

For the exercise at hand, \( g(x) = x^{3/2} \), apply the power rule:

• Identify \( n \), which is \( 3/2 \).
• Add 1 to \( n \), giving \( 5/2 \).
• Express the integral as \( \frac{x^{5/2}}{5/2} + C \), simplify it to get \( \frac{2}{5} x^{5/2} + C \).

This simplified method makes integration of polynomial terms straightforward, allowing students to find antiderivatives easily.

Differentiation

Differentiation is the process of finding the derivative of a function. Derivatives tell us how a function changes at any given point, revealing the function's rate of change. In this exercise, differentiation helps verify the antiderivative.

Once you find the antiderivative \( G(x) \), which is \( \frac{2}{5} x^{5/2} + C \), you use differentiation to ensure that \( G'(x) = g(x) \). Differentiating \( G(x) \) involves applying the power rule in reverse:

• Multiply by the exponent: \( 5/2 \cdot \frac{2}{5} x^{5/2 - 1} \)
• This simplifies to \( x^{3/2} \).
• Since \( G'(x) = x^{3/2} \), it confirms the original function \( g(x) \), verifying that the antiderivative was correctly found.

Verification of Solution

Verification of a solution ensures that the function obtained from integration is correct. It involves differentiating the antiderivative to check if it results in the original function.

To verify:

• Differentiate the antiderivative \( G(x) = \frac{2}{5} x^{5/2} + C \).
• Using the power rule for differentiation, compute \( G'(x) \).
• Check the result: \( G'(x) \) should be \( x^{3/2} \), which matches the original function \( g(x) \).

Finding \( G'(x) = x^{3/2} \) confirms that our antiderivative is correct, thus ensuring the solution is solid and reliable. By tracing back steps through differentiation, we make certain of our integration work.